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ii^  2008  with  funding  from 

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http://www.archive.org/details/completearithmetOOpeckrich 


^rofesao-T'  of  Mathematics  and  A  at-ronomy  in  Columbia  College, 
and  of  Mechanics  in  the  School  of  Mines. 


-^r^TOflWMW 


NEWYORK,CHiaftG[0  AND  NewORIiEIMS. 


?FBLiSHERS'    NOTICE. 


PECK'S    MATHEMATICAL    SERIES. 


CONCISE,  CONSECUTIVE,  AND  COMPLETE. 


I.— First  Lessons  in  Numbers. 
II.— Manual  of  Practical  Arithmetic. 
III.— Complete  Arithmetic. 
VI.— Manual  of  Algebra. 

V. — Manual  of  Geometry  and  Conic  Sections. 
VI.— Treatise  on  Analytical  Geometry. 
VII.— Differential  and  Integral  Calculus. 
VIII.— Elementary  Mechanics  ^without  the  Calculus). 
IX.— Elements  of  Mechanics  (with  the  Calculus). 


'NoTE.^ Teachers  and  others  discovering  errors  in  any  of  the  above 
works  wiU  confer  a  famr  ly  communicating  tTiem  to  us. 


Copyright,  1877,  by  William  G.  Peck. 


PREFACE. 


rriHE  object  of  the  following  work  is  to  present,  in 
-^  logical  order  and  within  moderate  limits,  all  the 
fundamental  principles  of  arithmetic,  together  with  their 
most  important  applications  to  the  wants  of  the  student, 
the  artisan,  and  the  man  of  business. 

It  commences  with  the  simplest  elements,  and  pro- 
gresses by  natural  steps  to  the  highest  and  most  complex 
operations.  All  superfluous  matter  has  been  omitted, 
and  great  care  has  been  exercised  to  avoid  needless  mul- 
tiplicity of  cases  and  rules;  but  in  no  instance  has  any 
essential  principle  been  omitted  or  unnecessarily  ab- 
breviated. 

It  is  believed  that  the  definitions  are  plain  and  concise ; 
that  the  principles  are  stated  clearly  and  accurately ;  that 
the  demonstrations  are  full  and  complete ;  that  the  rules 
are  perspicuous  and  comprehensive ;  and  finally,  that  every 
branch  of  the  subject  is  amply  illustrated  by  well-graded 
examples  and  problems. 

The  order  of  logical  development  is  thought  to  be 
simple  and  practical,  the  meth(^  of  treating  successive 


IV  PREFACE. 

subjects  being  uniform  and  essentially  as  follows :  1°.  All 
necessary  definitions  are  given ;  2°.  A  few  mental  exer- 
cises are  then  introduced,  being  so  worded  and  so  ar- 
ranged as  to  lead  the  pupil  to  a  knowledge  of  the 
fundamental  principles  of  the  subject  under  considera- 
tion; 3°.  The  principles  thus  developed  are  used  in 
demonstrating  the  required  rule  ;  and  4°,  The  rule  is  then 
illustrated  and  enforced  by  a  sufficient  number  of  graded 
examples  and  problems. 

The  Author  takes  this  opportunity  to  thank  the  many 
teachers  who  have  aided  him  by  valuable  suggestions  and 
criticisms. 

Columbia  College, 
Sept.  U,  1877. 


CONTENTS 


NOTATION  AND  NUMERA- 
TION, r^g^ 

Formation  of  Numbers 1 

Classification  of  Numbers 8 

Places  of  Fij?ures 9 

Orders  of  Units 9 

General  Principles 10 

Periods  of  Figures 12 

Roman  Notation 15 


FUNDAMENTAL     OPERA- 
TIONS. 

I    Addition H 

Explanation  of  Signs 18 

Principles  of  Addition 20 

Operation  of  Addition 20 

II.  Subtraction 28 

Explanation  of  Signs    29 

Principles  of  Subtraction 29 

Operation  of  Subtraction 30 

III.  MtlXTIFLICATION 37 

Sign  of  Multiplication 38 

Eflfect  of  annexing  Ciphers 41 

Principles  of  Multiplication 41 

Additional  Definitions 44 

IV.  Division 48 

Methods  of  indicating  Division  50 

Object  of  Division 51 

Principles  of  Division 51 

Short  Division 52 

Long  Division 55 

Contractions  In  Division 58 

V.  Factoring  AND  Cancelling...  65 

Principles  of  Factoring 66 

Operation  of  Factoring 66 

Cancellation 68 

Object  and  Principles  of  Can- 
cellation   68 


PAGE 

VI.  Greatest  Common  Divisor 
AND  Least  Common  Multi- 
ple   72 

Methods  and  Principles 73 

Method  by  Factors 73 

Additional  Principle 74 

Method  by  Continued  Division.  75 

Least  Common  Multiple 76 

Definitions 76 

Operation    of  Least  Common 

Multiple 77 

FRACTIONS. 

I.  Common  Fractions 80 

Reduction  of  Fractions 84 

Addition  of  Fractions 92 

Subtraction  of  Fractions 96 

Multiplication  of  Fractions  ....    99 

Division  of  Fractions 104 

Contractions  in  Multiplication 
and  Division 108 

II.  Decimal  Fractions. 
Decimals,    and    the    Decimal 

Point Ill 

Notation  of  Decimals Ill 

Numeration  of  Decimals 113 

Decimal  Currency 114 

ReductionofCommonFractions 

to  Decimals 115 

Approximate  Results 116 

Addition  of  Decimals 117 

Subtraction  of  Decimals 120 

Multiplication  of  Decimals 124 

Division  of  Decimals 128 

in.  Contractions  and  Business 

Operations 134 

Aliquot  Parts 134 

Bills  and  Accounts 137 

Balancing  Accounts 139 


71 


OONTEITTS. 


COMPOUND    NUMBERS. 

I.  Definitions  and  Tables.         page 

Scales  of  Compound  Numbers .  141 

Tables  of  Currency 142 

Tables  of  Weight 143 

Tables  of  Time 145 

Measures  of  Length 146 

Measures  of  Surface 147 

Measures  of  Volume  and  Ca- 
pacity   149 

Angular  Measure  &  Longitude.  151 
Metric  System 153 

II.  Reduction. 

Reduction  Descending 157 

Reduction  Ascending 160 

ni.  Addition  of  Compound  Num- 
bers    167 

IV.  Subtraction   op    Compound 

Numbers  ,  —  171 

V.  Multiplication  of  Compound 

Numbers 177 

VI.  Division  of  Compound  Num- 

bers    182 

Longitude  and  Time 188 

PERCENTAGE     AND     ITS 
APPLICATIONS. 

1.  Percentage 190 

Principles  of 192 

IT.  Commission 199 

III.  Insurance 203 

IV.  Profit  AND  Loss 206 

V.  Taxes 209 

On  Property  and  Polls 209 

Method  of  laying  a  Tax 210 

VI.  Simple  Interest 211 

Annual  Interest 222 

Notes 223 

Partial  Payments 223 

Methods  of  Settlement 224 

Supreme  Court  Rule 224 

Mercantile  Rule  226 

VII.  Compound  Interest 228 

VIII.  Discount 

Commercial  Discount 231 

Present  Value  and  True  Dis- 
count   232 

Banks  and  Bank  Discount 234 

Method  of  Discounting  a  Note.  235 

IX.  Stocks  and  Bonds 238 

United  States  Bonds 239 


PAGX 

X.  Exchange 242 

Drafts 242 

Acceptances 243 

Domestic  Exchange 244 

Foreign  Exchange 246 

XI.  Equation  of  Payments 248 

Equation  of  Accounts 253 

Cash  and  Interest  Balance 255 

XII.  Custom  House  Business —    256 

RATIO  AND  PROPORTION. 

L  Ratio 259 

Methods  of  Expressing 260 

II.  Proportion 261 

Solution.    Principles  used 262 

Rule  of  Three 263 

Distributive    Proportion    and 

Partnership 268 

Analysis 269 

POWERS,       ROOTS,      AND 
PROGRESSIONS. 

L  Powers  275 

Involution 275 

n.  Roots 276 

Square  Root 276 

Cube  Root 280 

ni.  Progressions 282 

Arithmetical  Progression 282 

Geometrical  Progression 285 

MENSURATION. 


Polygons 

Triangles 

Property  of  Right-angled  Tri- 
angles  

Length  of  a  Circumference 

Area  of  a  Triangle 

Area  of  a  Parallelogram 

Area  of  a  Trapezoid 

Areaofa  Circle 

Surface  of  a  Sphere 

Volume,  or  Content,  etc 

Content  of  a  Pyramid 

Content  of  a  Sphere 

Board  Measure 

Timber  Measure ■ 

Method  of  Duodecimals 


291 
293 
293 
294 
294 
295 
296 
296 
297 


301 


MISCELLANEOUS     EX  A  31- 

PLES 306 


DEFINITIONS. 

1,  A  Unit  is  a  single   thing  ;    as,  one 
pound,  one  foot,  one  day. 

2.  A  Number  is  a  unit,  or  a  collection 
of  units;  as,  one  pound,  three  days,  five  feet. 

3.  Arithmetic  is  the  science  of  numbers. 
It  treats  of  the  properties  and  relations  of  numbers,  and  of  the 
methods  of  computation  by  means  of  numbers. 

Note. — In  what  follows,  the  expressions  1°,  2°,  3°,  etc.,  are  read 
first,  second,  thirds  etc. 

FORMATION      OF    NUMBERS. 

4. — 1°.  Numbers  from  one  to  ten  are  formed  by  collect- 
ing simple  units,  or  ones. 

A  single  unit  is  called  one  ;  one  and  one  more  are  two  ; 
two  and  one  more  are  three  ;  three  and  one  more  are  four ; 
and  so  on,  to  ten. 

2°.  Numbers  between  ten  and  one  hundred  are  formed 
by  collecting  te?is  and  ones. 

One  ten  and  one  are  eleven  ;  one  te7i  and  two  are 
twelve ;  one  ten  and  three  are  thirteen  ;  and  so  on  to 
two  tens,  or  twenty.  Two  tens  and  07ie  are  twenty-one ; 
two  tens  and  ttvo  are  twenty-two ;  and  so  on  to  three 


8  KOTA^TIOK     AKD     NUMERATION'. 

,  ^  Jeiis  qx^thxTty,  „  Four  te7is  are  forty  ;  five  tens  are  fifty  ; 
"c'r-'araTd^©  (M  ikf  ■fe'/i^'tenSf  or  one  hundred.     The  interme- 
diate numbers  between  thirty  and  forty,  forty  and  fifty, 
and  so  on,  are  formed  in  the  same  manner  as  those  be- 
tween twenty  and  thirty. 

3°.  Numbers  between  one  hundred  and  one  thousand  are 
formed  by  collecting  hundreds,  tens,  and  ones;  numbers 
between  one  thousand  and  te7i  thousand  are  formed  by 
collecting  thousands,  hundreds,  tens,  and  ones  ;  and  so  on, 
indefinitely. 

Numbers  formed  by  collecting  ones,  in  the  manner  just  explained, 
are  called  integers  ;  they  are  also  called  integral,  or  whole  num- 
bers. 

CLASSIFICATION      OF    NUMBERS. 

5.  Numbers  are  divided  into  two  classes,  abstract  and 
deno7ni7iate. 

An  Abstract  Number  is  a  number  whose  unit  is  not 
named;  as, five,  seven,  eleve7i, 

A  Denominate  Number  is  a  number  whose  unit  is 
named ;  as,  three  pounds,  six  7niles,  seve7i  7nonths, 

Denominate  numbei's  are  sometimes  called  concrete  numbers. 

A  denominate  number  may  be  either  si7nple  or  compound. 

It  is  a  Simple  Number  when  all  the  units  of  the  col- 
lection are  of  the  same  name  or  denomination;  as,  eight 
yards,  eleven  ounces,  five  feet. 

It  is  a  Compound  Number  when  all  the  units  of  the 
collection  are  not  of  the  same  denomination  ;  ns,  tlweefeet 
mid  six  inches,  four  hows  and  twenty  mimites,  ttvo  pounds 
and  eleven  ounces. 

Note. — All  integers  are  simple  numbers. 


KOTATIOK     AND     NUMEKATION.  9 

NOTATION      AND      NUMERATION, 

6.  Notation  is  the  method  of  writing  numbers  by 
means  of  figures,  or  of  letters. 

Numeration  is  the  method  of  reading  written  numbers. 

Fl  GU  RES. 

7.  The  following  figures  are  used  in  the  common,  or 
decimal  system  of  notation : 

0,       1,      2,      3,      4,      5,      6,      7,      8,      9. 

naught,     one,      two,     three,     four,     five,       bIx,     seven,    eight,    nine. 

These  figures,  taken  separately,  are  called  digits.     The 

first  one,  named  naught,  is  also  called  a  cipher,  or  zero  ; 

it  stands  for  no  number.     The  remaining  ones  are  called 

significant  figures  ;  they  stand  for  the  numbers  written 

below  them. 

Figures  are  not  numbers,  but  it  will  often  be  convenient  to  speak 
of  them  as  such  ;  in  these  cases,  it  is  to  be  understood  that  we  refer 
to  the  numbers  which  the  figures  represent. 

PLACES     OF     FIGU  RES. 

8.  If  several  figures  are  written  in  a  line,  the  one  on  the 
right  is  said  to  stand  in  the  first  place,  the  one  next  to 
the  right  stands  in  the  second  place,  the  one  next  to  it 
in  the  third  place,  and  so  on.  Thus,  in  the  expression 
3784,  4  stands  in  the  first  place,  8  in  the  second  place >  7 
in  the  third  place,  and  3  in  ihQ  fourth  place, 

ORDERS     OF     UN  ITS. 

9.  The  number  one  is  called  a  unit  of  the  first  order  ; 
the  number  ten,  regarded  as  a  collectictn  of  ones,  is  called 
a  unit  of  the  second  order  ;  one  hundred,  regarded  as 
a  collection  of  tens,  is  called  a  unit  of  the  third  order ; 


10  NOTATIOIT     AKD     NUMERATION. 

and  so  on  indefinitely,  the  unit  of  each  succeeding  order 
being  ten  times  that  of  the  next  lower  one. 

The  unit  of  the  fourth  order  is  one  thousand ;  the 
unit  of  the  fifth  order  is  ten  thousand  ;  the  unit  of  the 
sixth  order  is  one  hundred  thousand  ;  the  unit  of  the 
seventh  order  is  one  million  ;  and  so  on. 

Units  whose  order  is  not  named  are  supposed  to  be  units  of  the 
first  order,  or  ones. 

GENERAL    PRINCIPLES    OF    NOTATION    AND    NUMERATION. 

10.  Numbers  are  written  and  read  in  accordance  with 
the  following  principles : 

1°.  The  same  digit  always  represents  the  same  number 
of  units. 

2°.  The  order  of  units  represented  is  denoted  hy  the  place 
in  tohich  the  digit  stands. 

Z°.  A  cipher  standing  in  any  place  shows  that  the  num- 
ber contains  no  units  of  that  order. 

Thus,  the  expression  777  denotes  7  hundreds,  7  tens, 
and  7  units,  that  is,  it  stands  for  seve7i  hundred  and  seventy- 
seven.  In  like  manner,  the  expression  507  denotes  5  hun- 
dreds, 0  tens,  and  7  units ^  that  is,  it  stands  iorfive  hundred 
and  seven.  The  expression  240  stands  for  two  hundred 
and  forty. 

Note. — Places  of  figures  and  orders  of  units  are  counted  from 
right  to  left,  but  numbers  are  written  and  read  from  left  to  right. 

EXAMPLES     IN     NOTATION     AND     NUMERATION. 

11.  Any  number  less  than  one  thousand  may  be  written 

by  the  following 

RULE. 

Begin  at  the  left  and  write  the  figures  that  denote 

the  hundreds,  tens,  and  units,  in  their  proper  order. 


ITOTATIOK     A^D     KUMERATIOK.  11 

E55:amples. 
Write  the  following  numbers  : 

1.  Fifty  seven.  '  Ans.  57. 

2.  Ninety  four.  Ans.  94. 

3.  One  hundred  and  sixty  nine.  Ans,  169. 

4.  Nine  hundred  and  fourteen.  A7is.  914. 

5.  Three  hundred  and  sixty.  15.  Two  hundred  and  nine. 

6.  Two  hundred  and  seven.  16.  Five  hundred  and  fifty. 

7.  Nine  hundred  and  eight.  17.  Six  hundred  and  nine. 

8.  Seven  hundred  and  seven.  18.  Ninety  seven. 

9.  Nine  hundred  and  ninety.  19.  One  hundred  and  ten. 

10.  One  hundred  and  twelve.  20.  Two  hundred  and  six. 

11.  Three  hundred  and  four.  21.  Six  hundred  and  sixty. 

12.  Eight  hundred  and  sixty.  22.  Four  hundred  and  five. 

13.  Eight  hundred  and  four.    23.  Seven  hundred  and  six. 

14.  Three  hundred  and  eight.  24.  Six  hundred  and  seven. 
Any  written  number  less  than  one  thouspnd  may  be  read 

by  the  following 

RULE. 

Begin  at  the  left  and  read  the  hundreds,  tens,  and 

units,  in  their  order,  translating  figures  into  words. 

EXAMPLES. 

Read  the  following  numbers : 

1.  29.  Ans.  Twenty  nine. 

2.  107.         Ans.  One  hundred  and  seven. 

3.  118.         Ans.  One  hundred  and  eighteen. 

4.  506.  8.  270.  12.  186.  16.  999. 

5.  670.  9.  809.  13.  204.  17.  400. 

6.  977.  10.  422.  14.  309.  18.  207. 

7.  835.  11.  109.  15.  470.  19.  554. 


12  KOTATION     AKD     NUMERATION. 

Note. — Before  reading  a  number,  let  the  pupil  name  the  unit  of 
each  order  of  figures,  beginning  at  the  right ;  thus,  uniiSy  tens,  hun- 
dreds. ♦ 

From  what  precedes,  we  see  that  notation  is  the  operation  of 
translating  numbers  from  words  into  figures,  and  that  numeration  is 
the  operation  of  translating  numbers  from  figures  into  words. 

The  digits  of  a  number  indicate  natural  parts  into  which  it  may 
be  separated.  Thus,  986  may  be  separated  into  the  three  parts,  900, 
80,  and  6,  each  of  which  has  its  own  unit. 

PERIODS     OF     FIGURES. 

13.  Written  numbers,  containing  more  than  three  fig- 
ures, are  separated  into  periods  of  three  figures  each,  be- 
ginning at  the  right  hand;  the  left-hand  period  may 
contain  less  than  three  figures. 

The  first  period,  counting  from  the  right,  is  called  the 
period  of  units;  the  second  is  called  the  period  of 
thousands  ;  the  third  is  called  the  period  of  millions  ; 
and  so  on,  as  shown  in  the  following  table,  called 

THE     NUMERATION     TABLE. 

^Hods\      Trillions,    Billions,     Millions,   Thousands,    Units. 

«M  «1H  «M  <♦-( 

o  o  o  o 

02  QQ  CQ  CC  DQ 

•13  ti.i'O  t»_ind  ci_iT3  54_('^ 

g«H*S  g'HO  g^O  g-gO  2 

'g®5     'gS^     '§"■2     %    ^   ^      a«^^ 

Periods.  .  .    3    i    7?    8    3    2,    4    i    5,    8    i    6,    7    8    3 
The  number  written  above  is  read,  317  trillions,  832 
billions,  415  millions,  816  tlioiisands,  783. 

The  unit  of  the  first  period  is  the  simple  unit  one  ;  the 
unit  of  the  second  period  is  one  thousand  ;  the  unit  of 
the  third  period  is  a  thousand  times  one  thousand,  or  one 
million  ;  and  so  on,  as  indicated  in  the  table,  the  unit  of 


NOTATION     AKD     KUMERATION.  13 

each  period  being  equal  to  a  thousand  times  that  of  the 
next  lower  one. 

The  table  may  be  continued  to  any  desired  extent ;  the 
units  of  the  next  succeeding  periods  are  quadrillions, 
quintillions,  sextillions,  septillions,  octillions,  etc. 

Every  period,  except  tlie  left-hand  one,  must  contain  three  figures, 
but  they  may  all  be  ciphers.  Periods  that  contain  three  figures  are 
said  to  be  complete. 

Periods  are  written  and  read  as  explained  in  the  last  article.  In 
writing,  we  make  them  all  complete,  except  the  one  on  the  left ;  in 
reading,  we  name  the  unit  of  each,  except  the  one  on  the  right. 

ADDITIONAL     EXAMPLES. 

13.  Any  number  whatever  may  be  written  by  the  fol- 
lowing 

RULE. 

Begin  at  the  left  and  write  each  period  in  order, 
separating  it  from  the  following  one  by  a  comma. 
Write  the  following  numbers : 

1.  Ten  thousand,  two  hundred  and  six.    Ans.  10,206. 

2.  One  hundred  and  fourteen  thousand,  eight  hundred 
and  seventy  nine.  Ans.  114,879. 

3.  Seven  hundred  and  fifty  thousand,  three  hundred 
and  eighty  nine.  Ans.  750,389. 

4.  Nine  hundred  thousand,  three  hundred  and  fifty. 

5.  Six  million,  one  hundred  and  sixty  nine  thousand, 
four  hundred  and  thirty  seven. 

6.  Seventy  six  million,  four  hundred  thousand,  one 
hundred. 

7.  22  billion,  103  million,  576  thousand,  102. 

8.  102  triUion,  125  milUon,  403. 

9.  8  trillion,  7  billion,  and  76. 


14  NOTATlOIf     AND     KUMERATION. 

10.  41  quadrillion,  817  trillion,  217  billion.      ^ 

11.  107  quintillion,  200  million,  757  thousand,  365. 

12.  14  billion,  74  million,  231  thousand,  and  5. 
Any  written  number  may  be  read  by  the  following 

RULE. 

I.  Begin  at  the  right  and  point  it  off  into  periods 
of  three  figures  each ;  the  left-hand  period  may  con- 
tain less  than  three  figures. 

II.  Begin  at  the  left  and  read  the  periods  in  their 
order,  naming  the  unit  of  each,  except  that  on  the 
right. 

Note. — After  the  number  is  pointed  oflf,  the  pupil  should  name 
each  period,  beginning  at  the  right ;  thus,  units,  thousands,  miUionSy 
UUions,  etc. 

EXAM  PLES. 

Eead  the  following  numbers  : 

1.  104217.  Arts.  One  hundred  and  four  thousand,  two 
hundred  and  seventeen. 

2.  2304516.  Ans.  Two  million,  three  hundred  and 
four  thousand,  five  hundred  and  sixteen. 

3.  1001010.    Ans,  One  million,  one  thousand,  and  ten. 

4.  825314715.    Ans.  825  million,  314  thousand,  715. 

5.  7416.  13.  6003021715. 

6.  23562.  14.  4785003298. 

7.  475437.  15.  12303492816. 

8.  284871.  16.  117723326419. 

9.  1284576.  17.  8843412956. 

10.  4534218.  18.  543521798612. 

11.  88334172.  19.  254321496. 

12.  24137652.  20.  1546973200849. 


NOTATION^     AND     NUMERATION. 


15 


ROMAN      N  OTATION. 

14.  Roman  Notation  is  the  method  of  expressing 
numbers  by  letters.  The  letters  used  and  the  values  they 
express  are  shown  below : 

Letters    ...    I,         V,         X,  L,  C,  D,  M. 

Values,  ...    1,         5,  10,        50,        100,       500,      1000. 

Other  numbers  are  expressed  by  combining  these  letters 
according  to  the  following  principles : 

1°.  If  a  letter  is  repeated,  the  number  that  it  denotes  is 
repeated. 

2°.  If  a  letter  that  denotes  a  less  number  is  written  after 
07ie  that  denotes  a  greater  nuinber,  the  value  of  the  latter  is 
increased  by  that  of  the  former. 

3°.  If  a  letter  that  deiiotes  a  less  number  is  written  before 
one  that  denotes  a  greater  number,  the  value  of  the  latter  is 
diminished  by  that  of  the  former. 

If  a  letter  that  denotes  a  less  number  is  written  between  two  that 
denote  greater  numbers,  it  diminishes  the  latter,  but  does  not  affect 
the  former. 


The  method  of  applying  these 

principles  is  shown  in  the 

follomng 

TABLE 

I    denotes  1 

XI    denotes  11 

XXX  denotes  30 

CCCC  denotes  400 

II         "       2 

XII 

•         13 

XL 

"      40 

D               "       500 

III       "       3 

XIII 

'       13 

L 

"  .  50 

DC             "       600 

I^       u       4 

XIV 

'       14 

LX 

-'       60 

DCC           "       700 

V         «'       5 

XV 

'       15 

LXX 

"      70 

DCCC        «       800 

VI       "       6 

XVT 

16 

LXXX 

"      80 

DCCCC      "       900 

VII      "       7 

XVII 

'       17 

XC 

"       90 

M               "     1000 

VIII    "       8 

XVIII 

•       18 

C 

"     100 

MM            "     2000 

IX       •'       9 

XIX 

'       19 

CC 

"     200 

MDQCCLXXV  de- 

X       -     10 

XX 

'       20 

CCC 

"     300 

notes  1875. 

16  NOTATION     AKD     NUMERATION. 

EXAM  PLES. 

Read  the  following  numbers : 

1.  XXXIX.  5.  MMDXXXIL 

2.  XCLVIII.  6.  DCOXLIII. 

3.  MDOXIX.  7.  DCCOCXC. 

4.  DCCLIX.  8.  CCCLXXXIII. 

Write  the  following  numbers  by  the  Koman  method : 
9.  42.  13.  2,940. 

10.  84.  14.  3,317. 

11.  119.  15.  2,150. 

12.  1,214.  16.  1,555. 

Note. — A  dash  over  a  number  written  in  Roman  numerals  in- 
creases the  number  1,000  times.     Thus,  XXX  stands  for  30,000. 

revie>a;'    questions. 

(1.)  What  is  a  unit?  Example  ?  (2.)  What  is  a  number?  Ex- 
ample? (3.)  What  is  arithmetic  ?  What  does  it  treat  of  ?  (4.)  Ex- 
plain the  formation  of  numbers  from  one  to  ten;  from  ten  to  one 
hundred,  etc.  What  is  an  integer,  or  whole  number  ?  (5.)  How  are 
numbers  classified ?  What  is  an  abstract  number?  A  denominate 
number  ?  When  is  a  denominate  number  simple  ?  When  compound  ? 
(6.)  Define  notation.  Numeration.  (7.)  Name  the  ten  digits.  What 
other  names  has  the  figure  naught  ?  Which  are  significant  figures  ? 
(8.)  Explain  what  is  meant  by  the  place  of  a  figure  ?  (9.)  Explain 
what  is  meant  by  orders  of  units,  and  give  the  names  of  the  orders 
up  to  the  seventh.  (lO.)  What  are  the  three  principles  of  decimal 
notation  ?  How  are  places  and  orders  counted,  and  how  are  numbers 
written  and  read  ?  (11.)  Give  the  rules  for  writing  and  reading  any 
number  less  than  1000.  '(l^*)  What  are  periods  of  figures?  Name 
the  units  of  the  first  six  periods.  (13. )  Give  the  general  rules  for 
notation  and  numeration.  (14.)  What  is  Roman  notation?  Explain 
fche  method  of  writing  numbers  in  this  system. 


I.    ADDITION. 

DEFINITIONS. 

15.  Addition  is  the  operation  of  find- 
ing the  sum  of  two  or  more  numbers. 

16.  The  Sum  of  two  or  more  numbers 
is  a  number  that  contains  as  many  units  as  the  given  num- 
bers taken  together.  Thus,  5  days  is  the  sum  of  3  days 
and  2  days. 

The  numbers  added,  and  their  sum,  must  be  similar. 

17.  Similar  Numbers  are  those  that  have  the  same 

unit.    Thus,  3  yards  and  7  yards  are  similar,  but  3  yards 

and  7  days  are  not  similar. 

Abstract  numbers  are  always  similar,  because  they  have  the  same 
unit. 

MENTAL     EXERCISES. 

1.  John  has  4  apples  and  James  has  5  apples;  how  many 
have  both  ?  How  many  apples  are  4  apples  and  5  apples  f 
How  many  are  4  and  5  ?    5  and  4  ? 

2.  Frank  had  8  marUes  and  Peter  gave  him  7  more ; 
how  many  had  he  then  ?  What  is  the  sum  of  8  marUes 
and  7  marbles?    What  is  the  sum  of  8  and  7?  of  7  and  8  ? 

3.  An  arithmetic  class  consists  of  5  loys  and  7^*Vk; 


18  SIMPLE  NUMBERS. 

how  many  pupils  are  there  in  the  class  ?    How  many  are 

5  and  7?     7  and  5  ? 

Explanation. — Because  both  "boys  and  girls  are  pupils,  the  num- 
bers to  be  added  are  similar,  although  they  appear  to  be  dissimilar. 

4.  What  is  the  sum  of  5  dollars,  6  dollars,  and  9  dollars? 
What  is  the  sum  of  5,  6,  and  9  ?  of  6,  9,  and  5  ?  of  9,  6, 
and  5  ? 

5.  A  farmer  has  6  oxen,  9  cows,  and  8  calves;  how  many 
cattle  has  he  ?    How  many  are  6,  9,  and  8  ? 

6.  What  is  the  sum  of  10, 12,  8,  and  4  ?  of  4,  8, 10,  and 
12? 

EXPLANATION     OF     SIGNS. 

18.  The  sign  of  addition,  +?  is  called  plus ;  when 
placed  between  two  numbers,  it  shows  that  the  second  is 
to  be  added  to  the  first.  Thus,  the  expression  4  +  5  shows 
that  5  is  to  be  added  to  4. 

The  sign  of  equality,  =,  indicates  that  the  expres- 
sions between  which  it  is  placed  are  equal  to  each  other. 
Thus,  4  +  5  =  9,  indicates  that  the  sum  of  4  and  5  is 
equal  to  9. 

An  expression  of  equality  between  numbers  is  called  an 
Equation ;  the  part  on  the  left  of  the  sign  of  equality  is 
the  first  member  and  the  part  on  the  right  is  the  second 
member.  Thus,  in  the  equation  9  +  8  =  17  the  part 
9  +  8  is  the  first  member  and  the  part  17  is  the  seco7id 
member, 

MENTAL     EXERCISBS. 

1.  4  +  8  +  7  =  how  many? 

Note. — Let  the  pupil  supply  the  second  member  and  then  read 
the  equation. 


ADDITION.  19 

2.  4  +  5  +  7  +  ^  +  1  =  how  many  ? 

3.  7  7nen  +  4  men  +  3  men  +  9  wew  =  how  many  men? 

4.  3  dollars  +  2  dollars  +  9  dollars  =  how  many  t?o^ 

5.  6  oxen  +  12  cozi;^  +  7  ca^ve5  =  how  many  cattle  ? 

6.  6  +  9  +  4  +  3  +  9  +  2  +  1  =  how  many  ? 

7.  4  +  9  +  5  +  6  +  3  +  4  +  8=? 

8.  The  sum  of  4,  9,  3,  2,  7,  6,  and  5  equals  how  many  ? 

9.  4  ?/«r^s  +  9  yards  +  11  ^/o^r^s  +  8  yards  =  ? 

Note. — The  signs  of  interrogation  in  examples  7  and  9  indicate 
that  the  second  members  are  to  be  supplied  by  the  pupil. 

10.  14.  ft.  +  dft.  +  7/^.  +  9/^.  +  10/^.  =  ? 

11.  9  +  9  +  7  +  3  +  6  +  8  +  7  +  4=? 

Let  the  pupil  add  the  following  columns: 
(12.)  (13.)  (14.)  {Ih.)  (16.)  (17.)  (18.)  (19.)  (20.)  (21.) 


6 

7 

3 

9 

5 

4 

7 

6 

7 

3 

7 

4 

8 

2 

1 

2 

8 

4 

2 

6 

8 

3 

2 

8 

8 

2 

7 

4 

4 

7 

5 

9 

8 

3 

2 

4 

6 

5 

3 

3 

4 

8 

3 

9 

9 

4 

2 

5 

5 

4 

2 

4 

5 

6 

3 

5 

3 

7 

7 

9 

3 

7 

4 

5 

7 

5 

4 

8 

2 

1 

8 

5 

9 

4 

6 

6 

1 

9 

9 

8 

1 

2 

6 

7 

4 

9 

5 

2 

6 

2 

Note. — The  operation  of  adding  a  column  of  figures  should  be 
abbreviated  by  simply  naming  the  result  of  each  step.  Thus,  in 
example  12,  the  pupil  should  say  1,  9,  13,  14, 18,  23,  31,  38,  44. 

The  exercise  may  be  varied  by  adding  each  column  from  top  to 
bottom;  also  by  adding  the  lines  horizontally  both  forward  and 
backward. 


20  SIMPLE   NUMBERS. 

PRINCIPLES     OF     ADDITION. 

19.  The  operation  of  addition  depends  on  the  follow- 
ing principles : 

1°.  Any  number  is  equal  to  the  sum  of  all  its  parts. 
2°.  The  sum  of  two  or  more  nu7nbers  is  equal  to  the  sum 
of  all  their  parts. 

OPERATION      OF     ADDITION. 

20.  Let  it  be  required  to  find  the  sum  of  564,  783,  and 
688. 

Explanation. — Having  written  the  numbers  so  that   Operation. 
units  of  the  same  order  stand  in  the  same  column,  we  564 

begin  at  the  right  and  add  each  column  separately.  733 

The  sum  of  8,  3,  and  4,  is  15  units,  that  is,  1  ten  and  5  nnn 

units  ;  we  write  the  5  in  the  column  of  units  and  carry       ■ 

forward  the  1  and  add  it  to  the  column  of  tens.  The  2035 
sum  of  the  tens,  thus  increased,  1  +  8  +  8  +  6,  is  23 
tens,  that  is,  2  hundreds  and  3  tens  ;  we  write  the  3  in  the  column  of 
tens  and  carryforward  the  2  and  add  it  to  the  column  of  hundreds. 
The  sum  of  the  hundreds,  thus  increased,  3  +  6  +  7  +  5,  is  20  hun- 
dreds, that  is,  2  thousands,  and  0  hundreds;  as  this  is  the  last  col- 
umn, we  set  down  the  entire  sum.  The  resulting  number,  2,035,  is 
the  required  sum,  because  it  is  the  sum  of  all  the  units,  tens,  and 
hundreds  of  the  given  numbers  (Art.  1.9). 

In  like  manner  other  numbers  may  be  added ;  hence,  we  have 
the  following 

RULE. 
/.   Write  the  nuwibers  so  that  units  of  the  same 
order  shall  stand  in  the  same  column. 

II.  Add  the  column  of  units ;  set  down  the  sim- 
ple units  of  the  sum,  and  if  there  are  any  tens, 
cannj  them  forward  and  add  them  to  the  next 
column, 

III.  Add  the  column,  of  tens  /  set  down  th^  simply 


ADDITION.  21 

tens,  and  if  there  are  any  hundreds,  cam^y  them 
forward  and  add  them  to  the  next  column. 

IV.  Continue  this  operation  till  all  the  columns 
have  been  added.  Set  down  the  entire  sum  of  the 
last  column. 

EXAMPLES. 

Perform  the  following  additions : 

(1.)      (2.)      (3.)  (4.) 

315       29       215  8261 

423      814       27  3042 

719      302      891  171 


Sum,  1457     1145      1133      11474 

Tlie  rule  holds  good  for  all  simple  numbers,  whether  abstract 
or  denominate. 

(5.)  (6.)  (7.)  (8.) 

451  feet.  365  days,  187  pounds.  124  things. 

817  feet.  821  days.  203  pounds.  287  things. 

302  feet.  900  days.  866  pounds.  59  things. 

917  feet.  76  days.  771  pounds.  803  things. 

2487  feet.  2162  days.  2027  pounds.  1273  things. 

Proof  of  Addition". — Perform,  the  opei^ation  hi/ 
commencing  at  the  top  of  each  column,  and  adding 
downward.     The  sum  should-  he  the  same  as  hefore. 

Note. — Every  operation  in  addition  should  be  proved. 


(9.) 

(10.) 

(11.) 

(12.) 

(13.) 

9,102 

8,760 

25,678 

87  feet. 

62,743 

479 

325 

3,002 

236  feet. 

4,321 

73 

512 

21,001 

1,443  feet. 

78,731 

810 

786 

715 

2,010  feet. 

1,239 

4,312 

1,420 

1,630 

7,818  feet. 

4,241 

23 

SIMPLE 

NUMBERS. 

(14.) 

(15.) 

(16.) 

(17.) 

(18.) 

27  yards. 

7,478  days. 

117,064 

2,571 

2,476 

135  yards. 

423  days. 

92,973 

1,701 

7,884 

7,271  yards. 

79  ^fl^«/s. 

827,569 

973 

3,349 

185  ?/fl!rcfe. 

8,102  tZ««/s. 

1,351 

2,045 

5,876 

19.  Add  7,384;  326;  6,780;  and  57.       Ans.  14,547. 

20.  Add  6,740;    9,745;    5,769;    8,031;   6,543;   2,052; 
and  9,999.  Ans.  48,879. 

21.  Add  89;  4,500;  423;  2,024;  5,408;  6'0,546;  9,401. 

22.  Add   83,746  2/«r^5  ;    ^,V(^  yards  ;    Q92,o7'7  yards  ; 
456  yards  ;  and  7  yards. 

23.  Add  935,473  ^o?Z«rs;   2G2  dollars  ;   13, S9'^  dollars  ; 

598,453  dollars  ;    25  dollars  ;    3,734  dollars  ;    and    72,405 

dollars. 

The  sign  $  written  before  a  number  signifies  dollars ;  thus,  the 
expression  $120  is  read  120  dollars. 

24.  Find  the  sum  of  $93,180;  $279;  $8,711;  $371,800; 
$65  ;  and  $212,818. 

25.  Add  3,415 ;   17,382 ;   81,845 ;   162,345 ;   and  8,342. 

26.  Add   8,492 /ee^;    U,692  feet  j    112,897  feet ;    and 
117,712  feet. 

27.  Add  $8,842;   $31,887;   $113,214;   and  $887,319. 

28.  Add  385,842;   112,817;   32,413;   and  33,335. 

29.  Add  $88,141;   $32,314;   $141,003;   and  $89,947. 

30.  Add  114,312;   87,808;   3,214;   896;   and  87. 

31.  Add  8,730 ;   3,021 ;   785 ;   879 ;   and  92. 

32.  Add  $87;  $78;   $114;   $289;   $176;   and   $95. 

33.  ^2,3Uyds.;   119,3^2  yds.;   S,9Q2  yds.  ;  8,962  yds. 

34.  Add  17,439;  410,864;   842,317;   345,876;   79,884; 
and  18,719. 


ADDITION.  23 

35.  Add  714,312;  182,416;  312,867;  382,843;   79,816 
and  43,115. 

36.  Find  the  sum  of  3,345,816;    2,882,314;    387,892 
4,381,500  ;  2,874,316  ;   and  887,342. 

37.  Add  188,841;  362,817;  411,217;  336,425;  814,316 
and  45,554. 

38.  Add  214,333;   286,329;  851,426;  303,249;  12,456 
17,324;   and  22,404. 

39.  Add    3,329,941 ;     187,693  ;     821,436  ;    2,227,438 
132,314 ;   and  283,304. 

40.  Add  193,391;  4,180,280;  7,814,312;  88,430;  92,872 
and  64,428. 

41.  Add     112,847;      186,320;      662,641;      3,400,300 
2,810,000;   and  749,209. 

42.  Add  682,817;  336,336;  4,150,209;  2,390,374;  and 
86,810,304. 

Dollars  and  cents  may  be  written  together,  the  cents  being  sep- 
arated from  the  dollars  by  a  point.  Thus,  the  expression  $17.84 
is  read  17  dollars  and  84  cents. 

Dollars  and  cents  may  be  added  like  simple  numbers.  In  writing 
them  down,  the  separating  points  must  stand  in  the  same  column. 


(43.) 

(44.) 

(45.) 

(46.) 

(47.) 

$18.73 

15.83 

1186.40 

1413.30 

$2,234.75 

23.47 

10.19 

75.75 

325.15 

3,821.62 

15.62 

27.03 

37.18 

414.82 

911.94 

7.91 

11.94 

201.92 

97.45 

89.69 

112.13 

203.07 

184.42 

111.32 

10,312.41 

648.21 

211.46 

36.35 

202.16 

9,102.70 

73.19 

305.24 

41.15 

113.27 

25,444.33 

19.06 

802.41 

72.27 

814.42 

42,829.77 

35.62 

111.37 

94.79 

316.81 

11,312.48 

24  SIMPLE   NUMBERS. 

48.  What  is  the  sum  of  $8,311.35, 127,494.62,  $143,596.22, 
$155,463.79,  $292,986.48,  $382,811.67,  $482,884.20,  and 
$919,902.20  ? 

49.  Add  $2,863,747.25,  $3,894,511.82,  $8,818,416.20, 
$215,714,381.46,  $747,719.87,  and  $59,107,411.28. 

50.  Find  the  sum  of  $53.42,  $881,16,  $416.49,  $1,381.40, 
$88.88,  $210.29,  $6.49,  $511.11,  $16.84,  and  $2,256.00. 

PRACTICAL     PROBLEMS. 

21.  A  Problem  is  a  question  proposed  that  requires 
an  answer.  The  operation  of  finding  the  answer  is  called 
the  solution  of  the  problem. 

Solve  the  following  problems  : 

1.  A  farmer  sold  a  span  of  horses  for  $318,  two  pairs 
of  oxen  for  $420,  and  six  cows  for  $290 ;  how  much  did 
he  receive  ?  A  7is.  $1,028. 

2.  A  man  bought  a  house  for  $24,500,  paid  $1,675  for 

repairs,  $3,140  for  furniture,  $375  for  taxes,  and  then  sold 

the  whole  for  $2,155  more  than  the  cost;   what  did  he 

receive?  Ans.  $31,845. 

Abbreviations. —  In  the  following  problems,  lbs.  stands  for 
pounds  ;  ft.  for  feet ;  yds.  for  yards ;  and  hu.  for  hunhels.  Other 
abbreviations  will  be  explained  in  their  proper  places. 

3.  A  wagon  is  loaded  with  5  boxes  of  goods ;  the  first 
weighs  473  Ihs.,  the  second  392  Ihs.,  the  third  479  Ihs.,  the 
fourth  1,217  Ihs.,  and  the  fifth  376  lbs. ;  what  is  the  weight 
of  the  entire  load  ?  Ans.  2,937  Ihs. 

4.  The  first  car  of  a  freight  train  contains  8,117  lbs.  the 
second  11,819  lbs.,  the  third  9,156  lbs.,  the  fourth  8,884 /i.9., 
the  fifth  10,398  lbs.,  and  the  sixth  9,982  lbs. ;  how  many 
pounds  are  there  in  all  ?  Ans.  58,356  lbs. 


ADDITION.  25 

5.  A  farm  contains  79  aci  es  of  woodland,  63  of  pasture 
land,  50  of  meadow  land,  and  73  of  arable  land;  how 
many  acres  in  the  farm  ?  Ans.  265  acres. 

6.  A  factory  turned  out  702  yds.  of  cloth  on  Monday, 
1,023  yds.  on  Tuesday,  1,107  yds.  on  Wednesday,  997  yds. 
on  Thursday,  910  yds.  on  Friday,  and  1,045  yds.  on  Satur- 
day ;  how  many  yards  did  it  turn  out  in  the  week  ? 

Ans.  5, '7S4:  yds. 

7.  A  merchant  owes  A.  $2,160,  B.  $3,879,  C.  $813,  D. 
$955,  and  E.  $1,796 ;  how  much  does  he  owe  in  all  ? 

Ans.    $9,603. 

8.  A  farmer  has  12  horses,  16  more  oxen  than  horses, 
42  more  cows  than  horses  and  oxen  together,  and  22  more 
calves  than  oxen  and  cows  together ;  how  many  in  all  ? 

9.  A  gentleman  built  a  house ;  the  carpenter  work  cost 
him  $4,285,  the  masonry  $3,950,  the  plumbing  and  grates 
$2,783,  the  painting  $1,975,  and  miscellaneous  work 
$3,992  ;  what  was  the  entire  cost  ? 

10.  A  merchant  buys  56,250  bu.  of  corn,  30,211  bu.  of 
oats,  18,312  bu.  of  barley,  2,197  bu.  of  wheat,  and  713  bti.  of 
rye ;  how  many  bushels  did  he  buy  altogether  ? 

11.  The  distance  from  Albany  to  New  York  is  144 
miles,  from  New  York  to  Philadelphia  90  miles,  from 
Philadelphia  to  Baltimore  98  miles,  from  Baltimore  to 
AVashington  38  7niles,  and  from  Washington  to  Norfolk  217 
7niles  ;  how  far  is  it  from  Albany  to  Norfolk  by  this  route  ? 

12.  In  a  lumber  yard  there  are  3 7,41 2 /if.  of  spruce, 
15,102//^.  of  pine,  9,187//.  of  oak,  171,812//f.  of  hemlock, 
7,413/if.  of  ash,  and  18,002/if.  of  chestnut;  how  many  feet 
are  there  of  all  kinds  ? 


26  SIMPLE  NUMBERS. 

13.  A  work  consists  of  6  yolumes;  the  first  volume 
contains  611  pages,  the  second  539,  the  third  687,  the 
fourth  599,  the  fifth  580,  and  the  sixth  679;  how  many 
pages  in  the  entire  work  ? 

14.  A  man  bequeaths  $15,750  to  his  daughter,  122,850 
to  each  of  two  sons,  and  twice  as  much  to  his  wife  as  to 
his  daughter;  what  is  the  amount  of  his  bequests  ? 

15.  The  population  of  Maine  is  627,413,  of  New  Hamp- 
shire 301,471,  of  Vermont  300,187,  of  Massachusetts 
1,240,499,  of  Connecticut  410,749,  and  of  Ehode  Island 
192,815;  what  is  the  aggregate  population  of  these  States? 

16.  In  1876  the  number  of  miles  of  railroad  in  the 
United  States  was  as  follows :  in  New  England  5,694,  in 
the  Middle  States  15,085,  in  the  Western  States  37,055, 
in  the  Southern  States  16,676,  and  in  the  Pacific  States 
2,960  ;  how  many  miles  in  all  ? 

17.  In  1876  the  popula^tion  of  the  several  divisions  of 
the  United  States  was  as  follows :  New  England  3,806,850, 
Middle  States  11,105,000,  Western  States  15,835,000, 
Southern  States  12,410,000,  Pacific  States  1,280,395;  what 
was  the  population  of  the  entire  country  ? 

18.  A  merchant  bought  parcels  of  cloth  containing 
respectively  3,912,  1,856,  2,011,  4,550,  937,  6,303,  1,856, 
2,024,  4,228,  1,345,  6,138,  607,  960,  2,445,  and  8,982  yards; 
how  many  yards  did  he  buy  in  all  ? 

19.  The  first  of  four  numbers  is  3,125,  the  second  is 
greater  than  the  first  by  5,108,  the  third  is  equal  to  the 
sum  of  the  first  and  second,  and  the  fourth  is  equal  to  the 
sum  of  the  third  and  first ;  what  is  the  sum  of  the  four 
numbers  ? 


ADDITIO:Nr.  27 

20.  The  ship  Orient  sailed  from  Marseilles  to  Buenos 
Ayres,  distant  6,375  miles,  thence  to  Valparaiso  2,764  miles, 
thence  to  San  Francisco  6,346  miles,  thence  to  the  Sand- 
which  Islands  2,152  miles,  thence  to  Melbourne  5,588  miles, 
thence  to  Yokohama  b,^%^  miles,  thence  to  Calcutta  5,115 
miles,  thence  to  Bombay  2,257  miles,  thence  to  Suez  2,006 
miles,  and  thence  back  to  Marseilles  1,^1^ miles;  what 
was  the  entire  distance  sailed  ? 

21.  A  merchant  commenced  business  with  the  following 
capital:  Cash  $18,471.25,  goods  worth  121,419.52,  bank 
stock  $7,418.00,  and  other  property  worth  14,314.17 ;  he 
gained  $12,315.42  the  first  year,  and  $11,124.86  the  second 
year ;  how  much  was  he  worth  at  the  end  of  the  second 
year? 

22.  An  agent  collected  from  different  individuals:  127.18, 
$32.52,  $41.70,  83.49,  $8.17,  $91.94,  $127.86,  $14.54,  $87-.78, 
$411.10,  and  $79.62  :  how  much  did  he  collect  in  all  ? 

23.  A  man  has  real  estate  worth  $20,114.50,  bank  stock 
worth  $15,779.82,  United  States  bonds  worth  $17,772.89, 
and  other  property  worth  $6,317.27  ;  what  is  the  value  of 
his  entire  property  ? 

24.  Find  the  sum  of  the  following  items  of  account : 
$21.27,  $49.18,  $412.25,  $44.74,  $86.92,  $311.10,  $8.14, 
$118.45,  $32.41,  and  $52.52. 

REVIEW     QUESTIONS. 

(15.)  What  is  addition  ?  (16.)  What  is  the  sum  of  two  or  more 
numbers?  Wliat  numbers  can  be  added?  (17.)  When  are  num- 
bers similar?  Illustrate.  (18.)  Explain  the  use  of  the  signs  of 
addition  and  of  equality,  (19.)  What  are  the  principles  of  ad- 
dition ?  (20.)  Give  the  rule  for  addition.  The  method  of  proving 
addition.     (21.)  What  is  a  problem  ?    The  solution  of  a  problem  ? 


28  SIMPLE     NUMBERS. 

II.      SUBTRACTION. 

DEFINITIONS. 

22.  Subtraction  is  the  operation  of  finding  the  dif- 
ference between  two  numbers. 

23.  The  Difference  between  two  numbers  is  a  num- 
ber which,  added  to  the  less,  will  produce  the  greater. 
Thus,  6  is  the  difference  between  10  and  4,  because 
4  +  6  =  10. 

The  greater  number  is  called  the  Minuend  ;  the  less 
number  is  called  the  Subtrahend  ;  and  their  difference 
is  called  the  Remainder. 

The  minuend,  the  subtrahend,  and  the  remainder  must  be  similar. 

MENTAL     EXERCISES. 

1.  James  has  9  marUes  and  Samuel  has  4  marbles  ;  how 
many  more  marbles  has  James  than  Samuel  ?  If  4  mar- 
bles are  taken  from  9  marbles,  how  many  will  be  left  ? 
4  from  9  leaves  how  many  ? 

2.  John  had  9  chestnuts  and  ate  6;  how  many  had  he 
left?  6  chestnuts  from  9  chestnuts  leayes  how  many 
chestnuts  9    6  from  9  leaves  how  many  ? 

3.  Henry  had  15  ce7its,  but  spent  9  cents ;  how  many 
cents  had  he  left  ?  9  from  15  leaves  how  many?  What  is 
the  difference  between  15  and  9  ?    16  and  9  ?    18  and  9  ? 

4.  William  had  14  apples,  of  which  he  ate  3  and  gave 
away  5  ;  how  many  had  he  then  ?  What  is  the  difference 
between  14  apples  and  the  sum  of  3  apples  and  5  apples  9 
8  from  14  leaves  how  many?  8  from  17  ?  8  from  19? 
What  is  the  difference  between  14  and  3  -f  5  ? 


SUBTRACTION.  29 

EXPLANATION     OF     SIGNS. 

34.  The  sign  of  subtraction,  — ,  is  called  minus  ; 
when  placed  between  two  numbers,  it  shows  that  the 
second  is  to  be  subtracted  from  the  first.  Thus,  5  —  3 
shows  that  3  is  to  be  subtracted  from  5. 

A  parenthesis,  ( ),  inclosing  two  or  more  numbers, 
shows  that  the  inclosed  expression  is  to  be  treated  as  a 
single  number.  Thus,  8  —  (5  —  3)  shows  that  the  differ- 
ence between  5  and  3  is  to  be  subtracted  from  8. 

MENTAL     EXERCISES. 

1.  What  is  the  difference  between  16  —  4  and  10  ? 

2.  What  is  the  difference  between  16  and  10  +  4  ? 

3.  20  sheep  +  2  sheep  —  (4  sheep  -f  7  sheep)  =  how 
many  sheep  f 

4.  19  lolls  —  (12  halls  —  6  halls)  =  how  many  halM 

5.  1 7  —  9  1=  how  many  ? 

6.  (17  -f  10)  -  (9  +  10)  =  how  many  ? 

7.  15  —  7  =  how  many  ? 

8.  (15  +  10)  -  (7  +  10)  =  how  many? 

9.  $15  —  $9  =  how  many  dollars? 

10.  $15  H-  $9  —  (13  +  $4)  =  ? 

11.  24  lbs,  -  (8  Tbs,  -  3  lbs>i  =  ? 

12.  (24  lbs.  +  10  lbs.)  —  (5  lbs.  -f  10  lbs.)  =  ? 

13.  (20  +  16)  -  (10  +  10  -  9)  =  ? 

14.  (30  -f  17)  —  (10  +  10  -  8)  =  ? 

15.  (40  +  15)  -  (30  +  7)  =  ? 

PRINCIPLES     OF     SUBTRACTION. 

25.  The  operation  of  subtraction  depends  on  the  fol- 
lowing principles ; 


30  SIMPLE     NUMBERS. 

1°.  If  all  the  parts  of  the  subtrahend  are  taken  from 
corresponding  parts  of  the  minuend,  the  sum  of  the  par- 
tial remainders  is  equal  to  the  required  remainder. 

2°.  If  the  same  number  is  added  to  both  minuend  and 
subtrahend,  their  difference  is  not  changed. 

OPERATION     OF     SUBTRACTION. 

26.  Let  it  be  required  to  find  the  difference  between 
565  and  393. 

Explanation.— We  write  the  subtra-  opebatiok. 

hend  under  the  minuend,  so  that  units  k^k 

of  the  same  order  shall  stand  in  the  same  ' 

column.     Then,   beginning  at  the   right  Subtrahend,  ^S 

hand,  we  see  that  3  units  from  5  units  Remainder      1'<'2 

leaves  2  units  ;  we  therefore  write  2  in  the 

line  below.  Because  9  tens  cannot  be  taken  from  6  tens,  we  increase 
the  latter  by  10  tens,  making  it  16  tens ;  now  9  tens  from  16  tens 
leaves  7  tens ;  we  therefore  write  7  in  the  line  below.  To  compen- 
sate for  the  10  tens,  or  1  hundred  added  to  the  minuend,  we  may 
diminish  the  5  hundreds  of  the  minuend  by  1  hundred,  or  what  is 
the  same  thing,  we  may  increase  the  3  hundreds  of  the  suhtrahend 
by  1  hundred  (Principle  2°),  which  gives  4  hundreds  ;  taking  4  hun- 
dreds from  5  hundreds,  we  have  1  hnndred,  which  we  write  in  the 
line  below.  The  resulting  number,  172,  is  the  sum  of  the  partial 
remainders  obtained  by  subtracting  the  parts  of  the  subtrahend 
from  corresponding  parts  of  the  minuend;  it  is,  therefore,  from 
Principle  1°,  the  required  remainder. 

In  like  manner  we  may  find  the  difference  between  any  two  num- 
bers ;  hence,  we  have  the  following 

RULE. 

I.  Write  the  less  number  under  the  greater,  so  that 
units  of  the  same  order  shall  stand  in  the  same 
column. 

II.  Beginning  at  the  right,  subtract  each  figure 
in  the  lower  line  from  the  one  dbove  it,  and  write  the 
difference  in  the  line  below. 


SUBTRACTION.  ^1 

///.  If  any  figure  in  the  lower  line  exceeds  the 
one  above  it,  increase  the  latter  hy  10,  perforin  the 
subtraction,  and  then  add  1  to  the  next  figure  in 
the  lower  line. 

The  operation  described  in  the  last  clause  of  the  preceding  rule 
is  called  carrying.  This  operation,  and  that  of  adding  10,  when  re- 
quired, are  performed  mentally. 


EXAM  PLES. 

(1.) 

(2.) 

(3.) 

(4.) 

From 

663 

976  Ihs, 

704 /if. 

1,806  yds- 

Subtract 

580 

531  Ihs. 

483/^. 

720  yds. 

Remainder, 

83 

445  lis. 

221ft. 

1,086  yds. 

(5.) 

(6.) 

(7.) 

(8.) 

Prom 

4,236 

80,502 

$46,095 

1555,555 

Subtract 

3,089 

38,672 

$28,736 

$123,456 

Remainder, 

1,147 

41,830 

$17,359 

$432,099 

Proof. — Add  the  remainder  to  the  subtrahend ;  if 
the  sum  is  equal  to  the  minuend,  the  work  is  correct. 

ILLUSTRATIONS. 

From  75,625  376,781  lbs.  $367,045  $84.16 

Subtract  24,319  95,845  lbs.  $106,253  $29.18 

Remainder,  51,306  280,936  Ibs.  $260,792  $54.98 

Proof,  75,625  376,781  lbs.  $367,045  $84.16 

What  is  the  difference  between 

13.  30,811  and  13,240?         18.  892,201  and  300,998? 

14.  27,880  and  9,226  ?  19.  900,000  and  233,333  ? 

15.  35,846  and  12,829  ?  20.  880,002  and  801,998  ? 

16.  75,901  and  17,980  ?  21.  900,892  and  395  ? 

17.  37,229  and  17,991  ?         22.  516,315  and  211,209? 


32  SIMPLE     NUMBERS. 

23.  100,304  and  62,818  ?        25.  758,901  and  349,806  ? 

24.  900,302  and  788,772  ?      26.  561,915,435  and  9,435  ? 

27.  The  sum  of  two  numbers  is  7,817,412,  and  one  of 
the  numbers  is  7,212,494 ;  what  is  the  other  number  ? 

28.  The  greater  of  two  numbers  is  230,011  and  the  less 
is  210,299  ;  what  is  their  difference  ? 

29.  The  sum  of  two  numbers  is  485,752,  and  the  less 
number  is  82,992  ;  what  is  the  greater  ? 

30.  What  is  the    difference    between    40,690,080    and 
699,090  ? 

Perform  the  following  indicated  subtractions : 

39.  $57,846,203  -  $7,756. 

40.  14,396,802  —  $83,846. 

41.  3,718,412  -  807,306. 

42.  887,892  —  709,378. 

35.  $814,316  -  $91,320.  43.  68,893  -  29,394. 

36.  $620,306  —  $413,314.  44.  4,924,863  —  43,989. 

37.  $813,864  —  $11,899.  45.  2,814,316  —  999,007. 

38.  41,336^6^5.—  7,814  «/flf5.  46.  8,904,306  —  304,216. 

47.  From  2,816,214/«f.  subtract  1,856,394 /if. 

48.  How  much  does  3,816,204  exceed  3,334,599  ? 

49.  What  is  the  difference  between  740,817  and  220,198  : 

50.  From  the  sum  of  862,141  and  32,843  subtract  884,109. 

51.  How  much  does  the   sum  of  39,418  and  27,362 
exceed  the  sum  of  19,823  and  29,819  ? 

52.  Find  the  sum  of  18,814  and  32,315,  and  subtract 
from  it  17,794. 

53.  From  the  sum  of  $8,833,  $141,209,  and  $11,362, 
subtract  the  sum  of  $2,843,  and  $10,906. 


31. 

81,423  -  20,120. 

32. 

80,200  —  1,875. 

33. 

18,714  —  13,392. 

34. 

123,387  —  94,816. 

SUBTRACTION.  33 

54.  From  the  sum  of  88,303  feet,  and  61,112  feet  sub- 
tract the  sum  of  74,395 /ee^,  and  0,202 /ee^. 

55.  From  105,242  +  522,801  subtract  131,1444-211,746. 

56.  From  $21.56  +  $42.87  +  $11.72  subtract  $48.99  -f- 
$2.65. 

57.  From  $2,117.24  subtract  $214.29  +  $119.94  +  11.88. 

58.  From  $38,140.20   subtract    $16,884.49  +  $22.27  + 
$46.71. 

59.  $4,547.18  +  $1,620.29  -  ($459.94  +  $100.87  +  $1,- 
257.00)  =z  ? 

60.  $88,641  +  $316.45  —  ($19,384.22  —  $6,211.88)  =  ? 

PRACTICAL      PROBLEMS. 

1.  A.  borrowed  of  B.  $9,780  and  paid  $2,176 ;  how  much 
remained  due  ?  Afis,  $7,604. 

2.  A.  purchased  a  farm  for  $10,000  and  paid  thereon 
$4,790  ;  how  much  remained  due  ?  Ans.  $5,210. 

3.  B.  bought  merchandise,  which  he  sold  for  $11,275, 
and  made  thereby  $2,114 ;  what  was  the  cost  price  ? 

Atis.  $9,161. 

4.  In  1860  the  population  of  Maine  was  627,413,  and  in 
1870  it  was  913,279  ;  what  was  the  gain  in  10  years  ? 

Ans.  285,866. 

5.  The  sum  of  two  numbers  is  9,427,  and  the  greater  is 
5,825  ;  what  is  the  less  number  ? 

6.  In  1790  the  population  of  Connecticut  was  238,141, 
and  in  1840  it  was  309,978 ;  what  was  the  gain  in  that 
period  ? 

7.  In  1840  the  population  of  Arkansas  was  97,574, 
which  was  a  gain  of  07,186  in  10  years;  what  was  the 
population  of  that  State  in  1830  ? 


34  •  SIMPLE     NUMBERS. 

8.  How  much  does  57,182  exceed  18,394  ? 

9.  A  merchant  commenced  business  with  a  capital  of 
$21,308,  and  retired  with  $74,114;  how  much  did  he 
make? 

10.  A.,  B.,  and  C,  commence  business;  A.  puts  in 
$35,000,  B.  $41,700,  and  C.  $36,150 ;  at  the  end  of  a  year 
they  have  together  $149,711 :  how  much  did  they  gain  ? 

11.  A  merchant  bought  500  yards  of  linen  for  $276, 
3,400  yards  of  muslin  for  $325,  and  75  yards  of  broad- 
cloth for  $318,  and  sold  the  whole  for  $1,316 ;  how  much 
did  he  gain  ? 

12.  A.  has  a  yearly  income  of  $12,000;  of  this  he 
spends  for  rent  $2,750,  for  taxes,  repairs,  and  insurance, 
$814,  for  clothing  $1,342,  for  household  expenses  16,211, 
and  the  remainder  he  distributes  in  charity :  how  much 
does  he  distribute  ? 

14.  B.  has  $12,311,  and  after  paying  his  debts  and  giving 
away  $2,108,  he  has  remaining  $8,199  ;  what  is  the  amount 
of  his  debts  ? 

14.  A  merchant  bought  cloth  for  $1,592,  silk  for  $1,274, 
laces  for  $818,  and  sold  the  cloth  for  $2,102,  the  silk  for 
$1,190,  and  the  laces  for  $969  ;  how  much  did  he  gain  ? 

15.  A  landholder  owned  1,875  acres  in  Illinois,  2,396 
acres  in  Indiana,  and  13,742  acres  in  Michigan ;  of  this 
lie  sold  813  acres  in  Illinois,  372  acres  in  Indiana,  and 
7,411  acres  in  Michigan  :  how  many  acres  has  he  remain- 
ing ? 

16.  A.,  B,,  and  C,  are  in  trade ;  A.  gains  $7,055,  B. 
gains  $813  less  than  A.,  and  C.  gains  as  much  as  A.  .and 
B.  together,  lacking  $994 :  what  do  they  all  gain  ? 


SUBTRACTION.  35 

17.  A.  bought  a  farm  for  18,192,  expended  $2,815  for 
improvements,  paid  $387  for  taxes,  and  then  sold  ifc  so  as 
to  lose  $2,282 ;  for  what  did  he  sell  it  ? 

18.  A  man  worth  $18,000  left  $4,287  to  his  elder  son, 
$3,751  to  his  younger  son,  $3,219  to  his  daughter,  and  the 
remainder  to  his  wife  ;  what  was  the  wife's  portion  ? 

19.  A  man  was  21  years  old  in  1843;  in  what  year  will 
he  be  75  years  old  ? 

20.  A  merchant  bought  4  cargoes  of  grain ;  the  first 
contained  6,705 J?^.,  the  second  contained  842^w.  less  than 
the  first,  the  tJiird  contained  911  Jm.  more  than  the  second, 
and  the  fourth  contained  3,092^2^.  less  than  the  second 
and  third  together :  how  many  bushels  were  there  in  the 
four  cargoes  ? 

21.  A  man  bought  three  estates ;  for  the  first  he  gave 
$5,260,  for  the  second  he  gave  $3,585,  and  for  the  third 
he  gave  as  much  as  for  the  first  two  together ;  he  after- 
ward sold  them  all  for  $15,280 :  did  he  gain  or  lose,  and 
how  much  ? 

22.  A.  travels  due  east  at  the  rate  of  19  miles  an  hour ; 
B.  starts  from  the  same  place  1  hour  later  and  travels  in 
the  same  direction  at  the  rate  of  13  miles  an  hour ;  how 
far  apart  are  they  3  hours  after  A.  starts  ? 

23.  A.  travels  due  north  at  the  rate  of  17  miles  an  hour; 
B.  starts  from  the  same  place  an  hour  earlier,  and  travels 
due  south  at  the  rate  of  11  miles  an  hour ;  how  far  apart 
are  they  4  hours  after  A.  starts  ? 

24.  A  merchant  commenced  business,  having  in  cash 
$4,152,17,  in  goods  $11,443.12,  and  in  other  property 
$5,794.22;  at  the  end  of  a  year  he  had  in  cash  $2,158.23, 


36  SIMPLE     KUMBERS. 

in   goods   117,411.98,  and  in  other  property  $6,239.14: 
how  much  did  he  gain  in  the  year  ? 

35.  A  gentleman  purchased  a  house  for  $12,873.75,  a 
carriage  for  1720.50,  a  span  of  horses  for  $591.45,  and  a 
saddle  horse  for  1212 ;25 ;  he  paid  for  them  at  one  time 
$4,374.16,  at  another  time  $3,495.17,  and  at  a  third  time 
$2,675.14 :  liow  much  remained  unpaid  ? 

26.  The  areas  of  the  New  England  States  are  as  follows : 
Maine  has  30,408  square  miles,  New  Hampshire  has  9,386, 
Vermont  9,420,  Massachusetts  7,845,  Connecticut  4,693, 
and  Rhode  Island  1,395 ;  how  many  fewer  square  miles 
has  Maine  than  all  the  rest  together  ? 

27.  A  drover  bought  24  oxen  for  $1,214.26,  42  cows  for 
$2,111.79,  and  40  calves  for  $397.11 ;  he  sold  the  oxen  for 
$1,519.45,  the  cows  for  $2,237.18,  and  the  calves  for 
$318.27  ;  what  did  he  gain  by  the  transaction  ? 

28.  America  was  discovered  in  1492,  which  was  128  years 
before  the  settlement  of  New  England ;  in  what  year  was 
New  England  settled  ? 

29.  A  man  having  a  sum  of  money,  earned  $8,211,  and 
afterward  lost  $2,114,  when  he  found  that  he  had  $11,415 ; 
how  much  had  he  at  first  ? 

30.  In  a  division  there  were  11,376  men,  of  whom  696 
were  killed  in  battle  ;  how  many  remained  ? 

REVIE^A/'      QUESTIONS. 

(22.)  What  is  subtraction  ?  (23.)  What  is  the  difference  between 
two  numbers  ?  Illustrate.  What  is  the  minuend  ?  The  subtrahend  1 
The  remainder?  (24.)  What  is  the  name  and  use  of  the  sign  of 
subtraction?  What  is  the  use  of  the  parenthesis?  (25.)  What 
are  the  principles  of  subtraction  ?  (20.)  Give  the  rule  for  subtrac- 
tion.    How  is  subtraction  proved  ? 


MULTIPLICATIO]Sr.  37 

III.     MULTIPLICATION. 

DEFINITIONS. 

27.  Multiplication  is  the  operation  of  taking  one 
number  as  many  times  as  there  are  units  in  another. 

The  first  number,  or  the  number  to  be  repeated,  is 
called  the  Multiplicand ;  the  second  number  is  called 
the  Multiplier ;  and  the  result  is  called  the  Product. 
Thus,  4  imiUiplied  by  3  is  equal  to  4  +  4 -f  4,  or  to  12. 
Here  4  is  the  multiplicand,  3  is  the  multiplier,  and  12  is 
their  product. 

Both  multiplicand  and  multiplier  are  called  Factors 
of  the  product.    Thus,  4  and  3  slyb  factors  of  12. 

MENTAL      EXERCISES. 

1.  What  is  the  cost  of  3  oranges  at  6  cents  apiece  ? 
Explanation. — Because  1  orange  costs  Gets.,  3  oranges  will  cost 

6  cts.  +  6  els.  +  Qcts.,  or  18  ds.  From  this  we  see  that  multiplication  is 
a  short  method  of  performing  the  operation  of  addition  when  the 
numbers  to  be  added  are  equal  to  each  other. 

2.  A  farmer  sells  6  calves  at  $9  each  ;  how  much  does 
he  receive  ?  $9  +  $9  +  $9  +  $9  +  $9  +  $9  =  ?  How  much 
is  6  times  19  ?     6  times  9  ? 

3.  If  a  man  earns  $5  a  day,  how  much  will  he  earn  in 
9  days  ?  What  is  9  times  $5  ?  What  is  the  product  of 
9  and  5  ? 

4.  How  many  are  4  times  5  ?     How  many  are  5  times  4  ? 
Explanation. — The  product  does  not  depend  on  *  *  #  * 

the  order  of  the  factors,  as  may  be  seen  in  the  diagram.  #  *  *  # 

If   we   take  the  stars  by  columns  we   have   4  times  *  *  *  * 

5  stars  ;  if  we  take  them  by  horizontal  rows,  we  have  *  #  *  # 

5  times  4  stars ;  in  either  case,  we  have  20  stars.  *  #  *  # 

5.  How  many  are  9  times  8  ?     8  times  9  ?     4  times  10  ? 

7  times  12?     12  times  12  ? 


38 


SIMPLE     NUMBERS. 


SIGN      OF      MULTIPLICATION. 

28.  The  Sign  of  Multiplication,  x,  when  placed 
between  two  numbers,  indicates  that  their  product  is  to  be 
taken.  Thus,  the  expression  5x7  shows  that  5  is  to  be 
multiplied  by  7,  or  that  7  is  to  be  7nultiplied  ly  5. 


CONDENSED      1 

VIULT 

IPLICATION 

TABLE. 

1 

2 

3|4 

5 

6 

7 

8 

9 

|10 

11 

il2 

2 

4 

6  1    8 

10 

12 

14 

16 

18 

20 

22 

I24 

3 

6 

9  i  12 

15 

18 

21 

24 

27 

30 

ZZ 

36 

4 

8 

12  1  16 

20 

24 

28 

32 

36 

40 

44 

48 

5 

10 

15  1  20 

25 

30 

35 

40 
48 

_45, 
54 

i5o 

i  60 

_5S„ 
66 

60 

6 

12 

18I24 

30 

36 

42 

72 

H 

14 

21     28 

35 

42 

49 

56 
64 

63 

72 

70 

77 

84 

8 

i6 

24I32 

40 

48 

56 

1 80 

88 

i  96 

9 

i8 

27  I36 

45 

54 

63 

72 

81 

i9o 

99 

|io8 

10 

20 

30  1  40 

50 

60 

70 

80 

90 

100 

no 

|l20 

11 

22 

33    44 

55 

66  1 

77 

88 

99 

|iio 

121 

1x32 

12 

24 

36I48 

60 

72  1 

84 

96 

108 

|l20 

132 

1 

1144 

13 

26 

39IS2 

65 

78  1 

91 

104 

117 

130 

143 

II56 

14 

28 

42  1  56 

70 

84  1 

98 

112 

126 

I14O 

154 

|i68 

15 

30 

45    60 

75 

^      1 
90  1 

105 

120 

135 

I15O 

165 

ii8o 

16 

32 

48    64 

80 

96 

1X2 

128 

144 

|i6o 

176 

I192 

17 

34 

51    68 

85 

102 

119 

136 

153 

!i7o 

187 

204 

18 

36 

54    72 

90 

108  i 

126 

144 

162 

|i8o 

198 

I216 

19 

38 

57r76| 

95 

114 

nz 

152  1 

171 

190 

209 

!228 

20 

40 

60    80 

100 

120 

140 

r6o  ! 

180 

200  1 

220 

{240 

Use  of  the  Table.— Find 
and  the  multiplier  in  the  first 


the  nuiltiplicand  in  the  upper  line 
column  ;  their  product  will  then  be 


MULTIPLICATION.  39 

found  in  the  same  column  with  the  multiplicand  and  in  the  same 
line  with  the  multiplier,  By  reversing  the  process,  any  number 
given  in  the  table  may  be  separated  into  factors.    Thus,  187=17  x  11. 

MENTAL      EXERCISES. 

1.  Add  by  2's  from  2  to  40.  By  3's  from  3  to  60.  By 
4's  from  4  to  80.    By  5's  from  5  to  100. 

2.  Subtract  by  2's  from  40  to  2.  By  3's  from  60  to  3. 
By  4's  from  80  to  4.     By  5's  from  100  to  5. 

Note. — Let  these  exercises  be  continued  to  the  limit  of  the  table. 

3.  What  is  the  cost  of  12  lbs.  of  sugar  at  15  cts.  a  pound  ? 
12  times  15  cts,  are  how  many  cents  ?  What  is  the 
product  of  15  ds.  by  12  ?    Of  15  by  12  ? 

4.  A  man  earns  $18  a  week,  and  it  costs  him  $11  a  week 
to  hve  ;  how  much  can^he  save  in  13  weeks  ?  in  15  weeks? 
in  9  weeks?     (118— $11)  xl3  =  ?    (18-11)  x  (14— 6)=:  ? 

Note. — The  multiplier  must  always  be  abstract ;  the  multipli- 
cand may  be  either  abstract,  or  denominate  ;  the  product  is  always 
similar  to  the  multiplicand.  In  practice,  we  multiply  as  though 
both  factors  were  abstract,  and  then  determine  the  unit  of  the  product 
from  the  nature  of  the  question. 

5.  (26-9)  X  8  =  ?  9.  8  X  10-14  =  ? 

6.  (24-13)  X  (15-7)  =  ?       10.  4  X  9—13  x  2  =  ? 

7.  (20-7)  X  (19-8)  =  ?        11.  (4  +  9)-^(18-7)  =  ? 


8.  (3  +  9)  X  (11  +  6)  =  ?  12.  (7  +  10)  X  (12-3) 


V 


MULTIPLICATION      BY     ONE     FIGURE. 

29.  Lfet  it  be  required  to  multiply  946  by  8. 

Explanation. — Multiplying  6  units  by  operation. 

8,  we  have  48  units;  that  is,  4  tens  and       Multiplicand,       946 
8  units ;  we  set  down  8  in  the  units'  place,        Multiplier  8 

and  carry  the  4  tens  to  the  next  column.  7" 

Multiplying  4  tens  by  8  we  have  32  tens,       Product,  7,568 

which  increased  by  the  4  tens  brought  forward  give   36  tens,  or 
3  hundreds  and  6  tens;  we  set  down  6  in  the  tens'  place,  and  carry 


40 


SIMPLE     NUMBERS. 


the  3  hundreds  to  the  next  column.  Multiplying  9  hundreds  by  8 
and  adding  the  3  hundreds  brought  forward,  we  have  75  hundreds, 
which  we  set  down.  The  resulting  number  7,568  is  the  required 
product. 

In  like  manner  we  may  proceed  in  all  similar  cases ;  hence,  the 
following 

RULE. 

Begin  at  the  right  hand  and  multiply  each  figure 
of  the  multiplicand  by  the  multiplier,  setting  down 
and  carrying  as  in  addition. 

Note. — We  may  multiply  by  any  number  from  10  to  20  by  the 
same  rule. 


EXAMPLES. 

Perform  the  following  multiplications: 

(1.)     - 

(2.) 

.      (3.) 

(4.) 

Multiplicand, 

357 

8645 

2079 

84123 

Multiplier, 

5 

8 

9 

6 

Product, 

1785              69160 

18711 

504738 

(5.) 

(6.) 

(7.) 

(8.) 

8842  yds. 

3749  in. 

13146  lbs. 

$81386 

4 

7 

9 

8 

85368  yds. 

26243  in. 

118314  lbs. 

1651088 

(9.) 

(10.) 

(11.) 

(12.) 

5G432 

13596 /i^. 

14382  lbs. 

$87645 

12 

15 

17 

19 

13.  43875x9  =  ? 

14.  14876//.  X 11  =  ? 

15.  $87653  X  14  =  ? 

16.  79792  ZZ^.v.  X 12  =  ? 

17.  16749  X  (24-7)  =? 


18.  86639  X  12  =  ? 

19.  ^39864  X  18  =  ? 

20.  $222794  x  19  =  ? 

21.  637489  x  9  =  ? 

22.  333333  x  16  =  ? 


MULTIPLICATION.  41 

EFFECT    OF    ANNEXING    CIPHERS. 

30.  Every  cipher  that  we  annex  to  a  number  moves 
each  of  its  digits  one  place  to  the  left,  that  is,  it  converts 
units  into  teyis,  tens  into  hundreds,  and  so  on ;  but  this  is 
the  same  as  multiplying  the  number  by  10  ;  hence, 

To  inultiply  a  numher  hy  10,  we  annex  one  cipher ; 
to  multiply  it  hy  100,  lue  annex  two  ciphers ;  to  mul- 
tiply it  hy  1000,  we  annex  three  ciphers ;  and  so  on. 

Thus,  75  X  10  =  750;  75  x  100  =z  7,500;  75  x  1000  = 
75,000 ;  75  x  10,000  =  750,000 ;  and  so  on. 

To  multiply  by  any  numher  of  tens,  we  first  multiply  by  the 
given  numher  and  then  annex  one  cipher  to  the  product ;  to  multiply 
by  any  nvmher  'of  hundreds,  we  multiply  by  the  given  numher  and 
annex  ^?C6>  ciphers  ;  to  multiply  by  any  number  of  thousands,  we  mul- 
tiply by  the  given  number  and  annex  three  ciphers  ;  and  so  on.  Thus, 
8x4  tens  —  32  tens  =  320  ;  6x4  hundreds  -  24  hundreds  =  2,400  : 
7  X  3,000  =  21,000  ;  and  so  on. 

PRINCIPLES    OF    MULTIPLICATION. 

31.  The  operation  of  multiplication  depends  on  princi- 
ples already  explained  and  also  on  the  following : 

If  all  the  imrts  of  the  multiplicand  are  multiplied  hy 
ea^h  part  of  the  multiplier,  the  sum  of  the  p^artial products 
is  equal  to  the  required  product. 

MULTIPLICATION     BY    ANY    NUMBER. 

32.  Let  it  be  required  to  find  the  product  of  458  and  346. 
Explanation. — Ha\ing  written  the 

numbers  so  that  units  of  the  same 
order  stand  in  the  same  column,  we 
begin  at  the  right  and  multiply  all 
the  parts  of  the  multiplicand  by  6, 
as  explained  in  Art.  29 ;  this  gives 
2748  for  the  first  partial  product. 

We  next  multiply  all  the  parts  of 
the  multiplicand    by  4  tens,   or  40.       Product,  158468 


OPERATION. 

Multiplicand, 

458 

Multiplier, 

346 

'     2748 

Partial  products, 

1833 

1374 

42 


SIMPLE     NUMBERS. 


Multiplying  8  units  by  40  (Art.  30),  we  have  320,  that  is,  3  hun- 
dreds and  2  tens ;  we  omit  the  cipher,  write  2  ^in  the  column  of 
tens,  and  carry  3  to  the  column  of  hundreds,  and  so  on  ;  this  gives 
the  second  partial  product. 

We  next  multiply  all  the  parts  of  the  multiplicand  by  3  hundreds. 
Multiplying  8  units  by  300  we  have  2400,  or  2  thousands  and  4  hun- 
dreds ;  we  omit  the  two  ciphers,  write  4  in  the  column  of  hundreds, 
and  carry  2  to  the  column  of  thousands,  and  so  on  ;  this  gives  the 
third  partial  product. 

The  sum  of  the  products  thus  obtained  is  158,468 ;  but  this  is 
the  sum  of  the  partial  products  found  hy  multiplying  all  the  parts  of 
the  multiplicand  hy  each  part  of  the  multiplier ;  it  is  therefore  the 
required  product  (Art.  31)- 

In  like  manner  we  may  find  the  product  of  any  two  numbers  ; 
hence,  the  following 

RULE. 

I.  Write  the  jnultipUer  under  the  multiplicand, 
so  that  units  of  the  same  order  shall  stand  in  the 
same  column. 

II.  Beginning  at  the  Hght,  multiply  all  the  parts 
of  the  multiplicand  hy  each  figure  of  the  multiplier, 
writing  the  first  figure  of  each  partial  product  under 
the  corresponding  multiplier. 

III.  Find  the  sum  of  the  partial  products. 


EXAMPLES 

(1.) 

(2.) 

(3.) 

(4.) 

(5.) 

843 

1817 

7287 

325 

9372 

27 

69 

75 

503 

98 

5901 

16353 

36435 

975 

74976 

1686 

10902 

51009 

1625 

84348 

22761 

125373 

546525 

163475 

918456 

Proof. — Multiply  the  multiplier  by  the  multipli- 
cand ;  if  the  product  is  the  same  as  before,  the  ivorJc 
is  correct. 


MULTIPLICATIOlf. 


43 


(6.)         Proof, 

(7.)         Proof, 

345            572 

835             794 

572            345 

794             835 

690          2860 

3340           3970 

2415          2288 

7515          2382 

1725           1716 

5845           6352 

197340       197340 

662990       662990 

Multiply 

8.  875  by  349. 

18.  46,137  by  841. 

9.  11,843  by  216. 

19.  50,246  by  322. 

10.  1,781  by  74. 

20.  61,532  by  742. 

11.  999  by  77. 

21.  184,387  by  994. 

12.  1,754  by  306. 

22.  97,418  by  887. 

13.  7,506  by  45. 

23.  107,309  by  1,206. 

14.  2,016  by  1,008. 

24.  320,009  by  344. 

15.  8,435  by  371. 

25.  99,897  by  284. 

16.  4,572  by  614. 

26.  44,479  by  3,227. 

17.  32,183  by  179. 

27.  56,263  by  7,777. 

When  the  multiplier  terminates  in  ciphers,  multiply   the   sig- 

nificant part,  and  annex  the  ciphers  as  explained  in  Article  30. 

(28.)              (29.)                (30.) 

(31.)               (32.) 

31                875                  42 

87                3194 

290                4300              31000             1500              127000 

8990          3762500          1302000         130500        405638000 

33.  87  by  78,000. 

36.  4,968  by  3100. 

34.  314  by  87,000. 

37.  19,872  by  26000. 

35.  414  by  82,000. 

38.  346,843  by  4500. 

If  both  multiplicand  and  multiplier  terminate  in  ciphers,  multi- 
ply the  significant  parts,  and  annex  as  many  ciphers  as  there  are 
in  both  factors. 


44  SIMPLE     NUMBERS. 

39.  8,840  by  7,250.  45.  32,400  by  32,400. 

40.  2,040  by  8,060.  46.  18,750  by  16,000. 

41.  10,800  by  870.  47.  37,590  by  92,000. 

42.  37,300  by  8,170.  48.  33,330  by  27,100. 

43.  88,320  by  36,000.  49.  877,000  by  209,000. 

44.  45,100  by  8,190.  50.  337,800  by  99,000. 

51.  3,876  X  879  -  2,799  x  74  zir  ? 

52.  12,483  X  4,520  —  38,795  x  89  =  ? 
63.  1,379  X  794  +  145,902  x  86  =  ? 

54.  (14,749  -  3,892)  x  (12,700  -  8,309)  =  ? 

55.  (265,484  —  142,184)  x  (8,794  —  3,684)  =  ? 

56.  (18,943  +  37,711)  x  (27,385  —  7,965)  =:  ? 

57.  (7,890  +  8,901)  x  (3,700  +  6,400)  =  ? 

58.  (276  +  3,276)  x  875  -  4,962  x  79  =  ? 

59.  (1,846  —  199)  x  79  —  (i;329  -  211)  x  9  =  ? 

60.  (5,946  +  9,544)  x  284  +  (8,305  +  95)  x  890  =  ? 

ADDITIONAL     DEFINITIONS. 

33.  A  Continued  Product  is  a  product  of  more 
than  two  factors.  Thus,  3  x  4  x  5  is  a  continued  product; 
it  indicates  that  the  product  of  3  and  4  is  to  be  multi- 
pHed  by  5. 

A  continued  product  may  contain  any  number  of  factors.  Its 
value  is  independent  of  the  order  of  the  factors.  Thus,  3x4x5  = 
3x5x4  =  4x3x5. 

34.  A  Composite  Number  is  a  number  composed 
of  two  or  more  integral  factors.  Thus,  21  is  a  composite 
number,  because  it  is  equal  to  3x7;  the  number  30  is 
composite,  because  it  is  equal  to  2  x  3  x  5. 

To  multiply  a  number  by  a  composite  number  whose 
factors  are  known  we  have  the  following 


multiplicatio:n^.  45 

RULE. 
Multiply  the  iiiultiplicand  hy  one  factor  of  the 
multiplier,  then  multiply  the  result  hy  another  factor, 
and  so  on,  till  all  the  factors  have  been  used. 

EXAMPLES. 

1.  Multiply  324  by  36,  that  is,  by  9  x  4,  or  by  VZ  x  3. 

riBST  OPERATION.  SECOND  OPERATION. 

324  324 

9  12 

2916      Partial  product.  3888      Partial  product. 

4  3 


11664      Total  product.  11664      Total  product. 

Note. — In  the  following  examples  the  factors  of  the  multiplier 
may  be  found  by  means  of  the  multiplication  table.  If  the  factors 
are  unequal,  we  generally  begin  with  the  greater  one. 

2.  873x144  =  ?  7.  736x48  =  ? 

3.  887/y.  X  84  =  ?  8.  4,315  x  176  =  ? 

4.  13,845x63=?  9.  $8,712x209  =  ? 

5.  38,257  yds.  x  96  =  ?  10,  48  x  11  x  15  x  16  =  ? 

6.  7,836/^5.  X  132  =?  11.  234x14x12x7  =  ? 

PRACTICAL    PROBLEMS. 

1.  What  will  455  lbs.  of  sugar  cost  at  14  cents  per 
pound  ?  Ans.  6370  ds.  =  $63.70. 

2.  What  will  692  pounds  of  beef  cost  at  26  cents  per 
pound  ?  Ans,  17,992  cts.  =  1179,92. 

3.  If  one  barrel  of  pork  costs  $15,  what  will  1,728 
barrels  cost?  ^?^5.  125,920. 

4.  If  a  train  travels  35  miles  an  hour,  how  far  will  it 
travel  in  425  hours?  Ans.  14,875  miles. 


46  SIMPLE  .  NUMBERS. 

5.  An  army  contains  106  regiments,  and  each  regiment 
contains  1,128  men  ;  how  many  men  in  the  army  ? 

6.  If  it  requires  720  barrels  of  provisions  to  feed  an 
army  for  one  day,  how  many  barrels  will  it  require  for  365 
days  ?  Ans.  262,800  barrels. 

7.  There  are  320  rods  in  a  mile ;  how  many  rods  are 
there  in  50  miles  ? 

8.  If  a  railway  costs  $42,500  per  mile,  and  is  385  miles 
long,  how  much  does  it  cost  ? 

9.  A  field  containing  56  acres  produces  29  iu.  of  rye  to 
the  acre ;  what  is  the  total  yield  '? 

10.  Sound  travels  at  the  rate  of  1,142  feet  per  second ; 
how  far  will  it  travel  3,600  seconds,  or  one  hour  ? 

11.  The  distance  from  New  York  to  Bridgeport  is  56 
miles,  and  there  are  320  rods  in  a  mile;  how  many  rods 
from  New  York  to  Bridgeport  ? 

12.  In  an  orchard  there  are  214  rows  of  trees,  and  each 
row  contains  241  trees ;  how  many  trees  are  there  in  the 
orchard  ? 

13.  What  is  the  continued  product  of  92,  37,  and  45  ? 

14.  A  freight  train  consists  of  21  cars;  each  car  contains 
85  barrels  of  flour,  and  each  barrel  of  flour  weighs  196 
pounds :  how  many  pounds  in  the  entire  cargo  ? 

15.  The  distance  from  Bridgeport  to  New  Haven  is  18 
miles;  each  mile  contains  1,760  yards,  and  each  yard  3 
feet :  how  many  feet  from  Bridgeport  to  New  Haven  ? 

16.  In  an  orchard  there  are  14  rows  of  peach  trees ; 
each  row  contains  27  trees,  and  each  tree  bears  108 
peaches :  how  many  peaches  in  the  orchard  ? 


MULTIPLICATION.  47 

17.  A  man  earns  $18.75  a  week,  and  all  his  expenses  are 
$11.25  a  week ;  how  much  can  he  save  in  37  weeks  ? 

18.  A  woman  bought  IS  yds.  of  ribbon  at  27  cts.  a  yard, 
and  42  yds.  of  muslin  at  1 6  cts.  a  yard ;  what  did  she  pay 
for  both  ? 

19.  A  man  has  a  barn  worth  $475,  a  house  worth  5  times 
as  much  as  the  barn,  and  land  worth  3  times  as  much  as 
the  house  and  barn  together;  what  are  they  all  worth  ? 

20.  A  man  traveled  295  miles  in  6  days ;  for  5  days  he 
traveled  at  the  rate  of  53  miles  a  day:  how  far  did  he 
travel  the  sixth  day? 

21.  The  diameter  of  Mercury  is  2,967  miles ;  the  diam- 
eter of  Saturn  is  24  times  that  of  Mercury ;  and  the  diam- 
eter of  the  sun  is  12  times  that  of  Saturn :  what  is  the 
sun's  diameter  ? 

22.  A  courier  had  to  travel  a  certain  distance  in  13 
hours ;  for  5  hours  he  traveled  at  the  rate  of  12  miles  an 
hour,  but  finding  that  he  was  behind  time,  he  increased  his 
speed  and  finished  the  journey  at  the  rate  of  14  miles  an 
hour :  what  was  the  distance  traveled  ? 

23.  The  distance  from  Chicago  to  Albany  is  835  miles; 
a  passenger  train  starts  from  Chicago  and  runs  toward 
Albany  at  the  rate  of  38  miles  an  hour,  and  at  the  same 
time  a  freight  train  starts  from  Albany  and  runs  toward 
Chicago  at  the  rate  of  13  miles  an  hour :  how  far  apart 
are  they  at  the  end  of  12  hours  ? 

24.  A  steamboat  runs  from  St.  Louis  to  Cairo  in  11 
hours  at  the  rate  of  17  miles  an  hour;  from  Cairo  to  Mem- 
phis in  15  Jirs.,  at  the  rate  of  16  ?m.  an  hour;  from  Memphis 
to  Vicksburgh  in  23  hrs.,  at  the  rate  of  18  mi.  an  hour;  and 


48  SIMPLE     NUMBERS. 

from  Vicksburgh  to  New  Orleans  in  21  hrs.,  at  the  rate  of 
19  mi.  an  hour :  what  was  her  running  distance  from  St. 
Louis  to  New  Orleans  ? 

25.  A  fox  starts  from  a  certain  place  and  runs  at  the 
rate  of  616  yds.  a  minute ;  at  tlie  end  of  three  minutes  a 
dog  starts  from  the  same  place  and  follows  the  fox  at  the 
rate  of  792  yds.  a  minute  :  how  far  apart  are  they  at  the 
end  of  9  minutes  ? 

REVIE^A^     QUESTIONS. 

(27.)  What  is  multiplication?  Multiplicand?  Multiplier?  Pro- 
duct ?  Define  factors.  (28.)  Explain  the  use  of  the  sign  of  multi- 
plication. (29.)  Give  the  rule  for  multiplying  by  a  single  figure. 
(30.)  How  do  you  multiply  a  number  by  10  ?  By  100  ?  By  1000  ? 
By  any  number  of  tens  ?  (31.)  What  is  the  fundamental  principle 
of  multiplication  ?  (32.)  Give  the  rule  for  multiplication  by  any 
number.  If  the  multiplicand  is  denominate,  what  will  be  the  nature 
of  the  product?  What  is  the  method  of  proving  multiplication? 
What  is  the  rule  for  multiplying  when  both  factors  terminate  in 
ciphers  ?  (33.)  What  is  a  continued  product  ?  How  many  factors 
may  such  a  product  contain ?  (34.)  What  is  a  composite  number? 
How  do  you  multiply  by  a  composite  number  whose  factors  are 
known  ? 


IV.     DIVISION. 

DEFINITIONS. 

35.  Division  is  the  operation  of  finding  how  many 
times  one  number  is  contained  in  another,  or  of  finding 
one  of  the  equal  parts  of  a  number. 

The  number  to  be  divided  is  the  Dividend  ;  the  num- 
ber by  which  it  is  divided  is  the  Divisor  ;  the  result  of 
the  division  is  the  Quotient ;  and  the  part  of  the  divi- 
dend that  remains  after  the  operation  is  the  Remainder. 


DIVISION.  49 

When  the  remainder  is  0,  the  division  is  said  to  be  exact  ;  in 
this  case  both  the  divisor  and  tke  quotient  are  Jactors  of  the  divi- 
dend. 

MENTAL      EXERCISES. 

1.  If  24  ajyples  are  divided  equally  among  6  boys,  how 
many  apples  will  each  boy  receive  ?  What  is  one  of  the 
6  equal  parts  of  24  apples  ?  How  many  times  is  6  con- 
tained in  24  ? 

Note. — To  use  the  multiplication  table  or  a  division  table,  find  the 
divisor  in  the  left  hand  column,  and  on  the  same  line  find  the  divi- 
dend ;  the  quotient  will  be  the  number  at  the  head  of  the  corres- 
ponding column.  Let  the  pupil  familiarize  himself  with  this  method 
of  using  the  table. 

2.  What  is  the  quotient  of  36  divided  by  6  ?  of  144 
by  12  ?  of  126  by  14  ?  of  72  by  9  ?  of  153  by  17  ?  of  90 
by  18?  of  95  by  19?  of  56  by  14? 

3.  How  many  7's  can  be  taken  from  23,  and  what  will 
remain  ?  What  is  the  quotient  of  23  divided  by  7,  and 
what  is  the  remainder  ?  Subtract  by  7's  from  23  as  far  as 
possible,  and  find  the  remainder. 

Note. — Division  may  be  regarded  ^s  a  short  method  of  continued 
subtraction.  The  number  of  times  that  the  divisor  can  be  taken  from 
the  dividend  is  equal  to  the  quotient. 

4.  If  47  peaches  are  divided  into  9  equal  parcels,  how 
many  will  there  be  in  each  parcel,  and  how  many  over  ? 
In  this  case,  what  is  the  dividend  9  The  divisor  9  The 
quotient  9    The  remainder  9 

5.  If  you  divide  126  by  11,  what  is  the  quotient,  and 
what  is  the  remainder?   149  by  12?   74  by  6  ?   190  by  15  ? 

6.  If  you  divide  $154  among  14  children,  how  much  will 
each  child  receive  ?  What  is  the  quotient  of  154  divided 
by  14  ?     How  many  times  is  14  contained  in  154. 


50  SIMPLE     NUMBERS. 

Note. — Division  is  performed  as  though  both  numbers  were 
abstract  and  the  unit  is  determined  from  tlie  nature  of  the  question. 
In  the  last  example  $1.54  is  equal  to  154:  cts.;  we  divide  154  by  14, 
which  gives  11  ;  hence,  the  answer  is  11  cts. 

If  the  dividend  and  divisor  are  similar  the  quotient  is  abstract ;  if 
the  dividend  is  denominate,  the  quotient  is  of  the  same  denomination 
as  the  dividend. 

7.  How  many  yards  of  cloth,  at  17  a  yard,  can  be  bought 
for  184  ?  If  12  yds.  of  cloth  cost  $84,  what  is  the  cost  of 
1  yd.  f  How  many  times  is  7  contained  in  84  ?  What  is 
the  quotient  of  84  by  7  ?   of  84  by  12  ? 

METHODS    OP     INDICATING    DIVISION. 

36.-1°.  The  sign  of  division,  -^,  when  placed  be- 
tween two  numbers,  indicates  that  the  first  is  to  be  divided 
by  the  second.  Thus,  the  expression  12  -^  3  indicates  that 
12  is  to  be  divided  by  3. 

2°.  The  same  operation  may  be  indicated  by  writing  the 
dividend  over  the  divisor,  with  a  line  between  them. 
Thus,  the  expression  -^^,  which  is  read  12  divided  by  3  is 
equivalent  to  the  expression  12  -r-  3. 

3°.  Division  may  also  be  indicated  by  writing  the  divisor 
on  the  left  of  the  dividend,  with  a  curved  line  between 
them.  Thus,  the  expression  3)12  is  equivalent  to  the 
expression  J^,  or  to  12  -^  3. 

MENTAL     EXERCISES. 

1.  What  is  the  quotient  of  144  by  12  ?  of  $96  ^  12? 
of  $96  -T-  $12  ?  of  84  -^  12?  How  many  times  is  $12 
contained  in  $84  ? 

2.  ¥  X  -V-  =  ?  4.  192  -^  (23  -  7)  =  ? 

3.  190  -^  19  -  54  -^  6  =  ?      5.  (200  _  8)  ^  16  =  ? 


DIVISION.  51 

6.  (108  +  45)  -^  (10  -f-  7)  =  ? 

7.  (4  +  9  +  26)  -^  13  =  ?     9.  (12  x  17)  -^  6  =  ? 

8.  156  -^  (19  -  6)  =  ?       10.  (12  X  16)  -^  (13  -  5)  =  ? 

11.  (99  -^  11)  X  (154  -^  14)  =  ? 

12.  The  product  of  two  numbers  is  64  and  one  of  the 
numbers  is  16 ;  what  is  the  other  ?  64  -f-  16  ==  how 
many  ? 

13.  Write  the  following  by  means  of  signs:  the  sum  of 

$12  and  $10,  divided  by  $11,  is  equal  to  the  quotient  of  the 

difference  between  $18  and  $4,  divided  by  $7. 

Note. — If  a  thing  is  divided  into  2  equal  parts,  each  part  is 
called  one  half;  if  divided  into  3  equal  parts,  each  is  called  one 
third  ;  if  into  4  each  is  one  fourth  ;  if  into  5  each  is  one  fifth  ;  and 
so  on.  Thus,  i  is  one  half;  f  is  one  half  of  3,  or  3  Iialves  of  1  ;  J  is 
ojie  third;  f  is  one  third  of  8,  or  two  thirds  of  1 ;  -|^  is  one  fourth;  f 
is  one  fourth  of  2,  or  2  fourths  oi  1  ;  |  is  one  fourth  of  3,  or  3  fourths 
of  1  ;  and  so  on.  Expressions  of  the  form  ^,  i,  |,  f ,  etc.,  are  called 
fractions.    Fractions  are  treated  more  fully  hereafter. 

14.  What  is  J  of  4  ?  J  of  8  ?  ^  of  16  ?  i  of  3  ?  |  of 
12  ?    i  of  15  ?    J  of  9  ?    J  of  4  ?    J  of  8  ?     i  of  04  ? 

15.  Read  the  expressions  f ,  f ,  |,  ^,  j\,  f|,  -^,  ||. 

OBJECT     OF      DIVISION. 

37.  Division  is  the  reverse  of  multiplication.    In 

multiplication,  we  have  two  factors  given,  to  find  their 
product ;  in  division,  we  have  the  2^'^oduct  and  one  factor 
given,  to  find  the  other  factor. 

PRINCIPLES     OF      DIVISION. 

38.  The  operation  of  division  depends  on  the  principles 
obtained  by  reversing  those  of  Article  30,  and  also  on  the 
following,  obtained  by  reversing  that  of  Article  31 : 

If  WG  divide  all  the  parts  of  the  dividend  by  the  divisor, 


52    '  SIMPLE   NUMBERS. 

the  sum  of  the  jmrtial  quotients  is  equal  to  the  required 
quotient, 

39.  There  are  two  cases:  1°.  Short  Division,  in 
which  the  divisor  contains  but  one  figure ;  and,  2°.  Long 
Division,  in  which  the  divisor  contains  more  than  one 
figure. 

In  the  first  case  most  of  the  operation  is  performed  mentally  ;  in 
the  second  case  the  different  steps  of  the  operation  are  written  out. 
The  principles  employed  are  the  same  in  both  cases. 

I°CASE.      SHORT     DIVISION. 

40.  Let  it  be  required  to  divide  26,812  by  4  : 

Explanation. — Having  written  the  operation. 

numbers   as  shown  in  the  margin,  we  Dividend, 

begin  at  the  left  and  divide  the  dif-  Divisor,       4  )  26812 

ferent  parts   of  the  dividend   by  the 
divisor.  Quotient,  6703 

Since  3  is  not  divisible  by  4,  we 
divide  26  by  4  ;  this  gives  6  for  a  quotient  with  2  for  a  remainder  -, 
hence,  there  are  6  thousands  in  the  quotient ;  we  write  6  in  the 
column  of  thousands  and  to  the  remainder  we  annex  the  following 
figure  of  the  dividend  giving  28  hundreds.  The  quotient  of  S8  by 
4  is  7  ;  hence,  there  are  7  hundreds  in  the  quotient ;  we  therefore 
write  7  in  the  column  of  hundreds.  Since  there  is  no  remainder  and 
since  1  is  smaller  than  4  there  are  no  tens  in  the  quotient  ;  we  there- 
fore write  0  in  the  place  of  tens  and  annex  the  following  figure  to  1 
giving  12  units.  The  quotient  of  12  by  4  is  3,  which  we  write  in  the 
column  of  units. 

In  like  manner  we  may  treat  all  similar  cases  ;  hence,  the  follow- 
ing 

RULE. 

/.  Write  they  divisor  on  tfie  left  of  the  dividend, 
and  draw  a  line  between  theiiv. 

II.  Divide  the  first  figure  of  the  dividend  by  the 
divisor  and  set  the  quotient  underneath,  or,  if  the 


Divisioif.  53 

first  figure  is  less  than  the  divisor^  divide  the  first 
two  figures  and  set  the  quotient  under  the  second. 

III.  To  the  remainder  annex  the  following  figure 
of  the  dividend,  divide  the  result  hy  the  divisor  and 
set  the  quotient  underneath,  or,  if  the  result  is  less 
than  the  divisor,  put  a  cipher  in  the  quotient,  annex 
another  figure,  and  proceed  as  he  fore. 

IV.  Continue  the  operation  till  all  the  figures  of 
the  quotient  have  been  found. 


EXAMPLES. 

Perform  the 

following 

divisions: 

(1.) 

(2.) 

(3.) 

(4.) 

(5.) 

5)785 

6)804 

8)1624 

7)392 

9)1926 

Ans,    157 

134 

203 

56 

214 

(6.) 

(7.) 

(8.) 

(9.) 

(10.) 

4)1544 

3)825 

8)4896 

9)792 

7)2415 

Ans.       380 

275 

612 

88 

345 

If  there  is  a  remainder  after  the  last  partial  division,  we  write 
it  over  the  divisor  and  annex  the  result  to  the  quotient.  Thus, 
27  -i-  4  =  6J  indicates  that  the  quotient  of  27  by  4  is  6  with  a 
remainder  3.  The  expression  6f  may  be  read  G  and  3  divided  hy  4, 
or  6  and  S/t^wr^As. 

(11.)  (13.) 

5)176  4)8140 


(13.) 

(14.) 

7)8146 

9)4023 

11634 

447 

Ans,        35|  2035 

Proof. — Multiply  the  quotient  hy  the  divisor  and. 
to  the  product  add.  the  remainder  ;  if  the  result  is 
equal  to  the  d^ividend,  the  worh  is  correct. 

Thus,  in  example  (11.),  35  x  5  +  1  =  176;  hence,  the 
work  is  correct. 


54 


SIMPLE     NUMBERS. 


Divide 

15.  12,360  by  4. 

16.  3,730  by  5. 

17.  20,202  by  6. 

18.  37,904  by  4. 

19.  90,872  by  8; 

20.  640,339  by  7. 


Divide 

21.  1639,145  by  7. 

22.  1454,396  by  $8. 

23.  321,314/Z>.<?.  by  7. 

24.  l,287,643//f.  by  9/^. 

25.  $21,416,314  by  9. 

26.  82,324,717i/f/5.  by8. 


We  may  divide  by  any  number  from  10  to  20  in  the  same  manner 
as  when  the  number  is  expressed  by  one  figure. 


(27.) 

10)8760 
876 


(28.) 
11)10978//. 
998//. 


(29.) 
12)8134 


(30.) 
11)203236  97??'. 
18476  ?w?:. 


Divide 

31.  1888,888  by  $12. 

32.  77,077  lbs.  by  11  lbs. 

33.  6,809?/c?.5.  by  112/^5. 

34.  $26,736  by  $12. 

35.  154,824  by  12. 

36.  14,784?/?!.  by  11m. 

43.  ($18,211  -  $13,179) 
,   44.  ($8,444  -  $4,514)  -^ 


677if 

Divide 

37.  137,418  by  12. 

38.  124,212  by  12. 

39.  89,580  by  15. 

40.  $24,311  by  16. 

41.  $33,244  by  $15 

42.  403,210  by  12. 
-V-  (19  _  14)  =  ? 

-  ($22  —  $15)  =  ? 


45.  ($53.92  -  $7.27)  -^  15  =  ? 

46.  ($211- $175)  -^-16=? 

47.  (2,500  lbs.  +  2,065  lbs.)  -^  11 

48.  ($100  +  $28.94)  -^  14  =  ? 

49.  ($100  —  $21.52)  -^  9  =  ? 

50.  {5,000yds.  -  ^20 yds.)  -^  10 

51.  ($76  -  $25.84)  -^  12  =  ? 

52.  (893//.  —  491//.)  -^  6  =  ? 


DIVISION*.  55 

2°    CASE.      LONG      DIVISION. 

41.   Let  it  be  required  to  divide  60,198  by  237. 

Explanation. — Having  written  tlie  operation. 

divisor   on   the   left,  we  divide  as  ex-       Divisor.  Dividend.  Quotiant. 
plained  below.  237  )  60198  (  254 

Since   60  is  not  divisible    by  337,  474 

tliere  are  no  thousands  in  the  quotient ; 
annexing   1,   we    have   601   hundreds.  !-«<» 

The  quotient  of  601  by  237  is  greater  1185 

than  2,  but  less  than  3  ;  hence,  there  oj^o 

are  2  hundreds  in  the  quotient ;  multi- 
plying  237  by  3  hundreds,  we  have  474  ^^" 

hundreds,  and  this  taken  from  601  hun-      Eemainder,      0 
dreds,  leaves  137  hundreds,  to  which 

we  bring  down  and  annex  the  9  tens,  giving  1,379  tens.  The  quo- 
tient of  1,379  by  337  is  greater  than  5  and  less  than  6  ;  hence,  there 
are  5  tens  in  the  quotient ;  multiplying  337  by  5  tens,  we  have  1,185 
tens,  and  this  taken  from  1,379  tens,  leaves  94  tens,  to  which  we 
bring  down  and  annex  the  8  units,  giving  948.  The  quotient  of  948 
by  337  is  4  ;  hence,  there  are  4  units  in  the  quotient ;  multiplying 
337  by  4,  we  have  948,  which  taken  from  948,  leaves  0. 

In  like  manner  we  may  treat  all  similar  cases  ;  hence,  the  fol- 
lowing 

RULE. 

/.  Find  how  many  times  the  divisor  is  contained 
in  the  fewest  possible  figures  on  the  left  of  the  divi- 
dend for  the  first  figure  of  the  quotient ;  multiply 
the  divisor  by  this  figure  and  subtract  the  product 
from  the  figures  used. 

II.  Annex  to  the  remainder  thus  found  the  next 
figure  of  the  dividend,  and  find  hoiv  mamj  times 
the  divisor  is  contained  in  the  result  for  the  second 
figure  of  the  quotient ;  jy%ultiply  the  divisor  by  this 
figure  and  subtract  as  before. 

III.  Continue  this  operation  till  all  the  figures  of 
the  quotient  have  been  found. 


56 


SIMPLE     K  UMBERS. 


In  applying  the  preceding  rule,  it  is  found  convenient  to  write 
the  cUcisor  on  the  left  and  the  quotient  on  the  right  of  the  dividend. 
Should  there  be  a  remainder  after  the  operation  is  completed,  it  is 
to  be  treated  as  explained  in  short  division. 

The  method  of  proof,  which  is  the  same  as  for  short  division,  is 
illustrated  in  the  example  below. 


EXAMPLES 

. 

Perform  the  following  divisions : 

1.  Divide  35,114  by  143. 

OPEBATIOK. 

Dividend. 

PROOF. 

Divisor,    143)35114(245    Quotient. 

245 

Quotient. 

286 

143 

Divisor. 

651 

735  ) 

572 

980 

-    Partial  products. 

794 

245      ' 

715 

79 

Remainder. 

79    Remainder. 

35114 

Dividend. 

The  practical  method  of  adding  the  remainder  to  the  product  of 
the  quotient  and  the  divisor  is  indicated  in  the  above  example. 


(2.) 
30)7452(207 

72 

(3.) 
124)373116(3009 
372 

(4.) 
37)11248(304 
111 

252 

1116 

148 

252 

1116 

148 

In  example  (3),  the  partial  dividend,  25,  is  smaller  than  the 
divisor  ;  we  therefore  write  0  for  the  second  figure  of  the  quotient 
and  bring  down  2,  the  next  figure  of  the  dividend,  and  proceed  as 
before. 

In  exami)le  (3),  both  11  and  111  are  smaller  tlian  the  divisor; 
we  therefore  write  two  ciphers  in  the  quotient,  bring  down  G,  and 
proceed  as  before. 


DIVISION.  57 

Divide  as  indicated : 

5.  7,812  -i-  36.  27.  $14,874  by  402. 

6.  16,758  -^  49.  28.  14,430  lbs.  by  74  lbs. 

7.  14,464  -r-  64.  29.  300,360  miles  by  120. 

8.  75,518  ^  718.  30.  61,712  horses  by  304. 

9.  40,698  -^-  399.  31.  $722,631  by  |91. 

10.  38,214  -f-  386.  32.  $317,094  by  $82. 

11.  51,171 -^  111.  .    33.  1,731,195//.  by  73//. 

12.  10,368  ^  144.  34.  7,318,080  by  55. 

13.  27,264  -^  96.  35.  76,131,702  by  46. 

14.  88,534  -^  184.  36.  31,231,737  by  37. 

15.  20,615  -^  95.  37.  13,261,467  by  381. 

16.  45,579  -f-  209.  ,  38.  1,281,524  by  761. 

17.  51,867  -V-  112.  39.  13,189,212  by  937. 

18.  309,927  -^  309.  40.  728,807  by  731. 

19.  765,870  -^  98.  41.  $1,477.35  -7-  45  =  ? 

20.  1,536  -^  16.  42.  947,387  -^  54  r=r  ? 

21.  2,304  lbs.  -^- 12.  43.  145,260  -^  108  =  ? 

22.  1,360  yds.  -r-  16  yds.  44.  14,420,946  -^  74  =  ? 

23.  47,708//.  -^  27//.  45.  $813,204.25  -^  25  =  ? 

24.  $71,556  -f-  201.  46.  $24,411.75  -^  75  =  ? 

25.  30,056  lbs.  -^  884  lbs.  47.  435,780  -^  216  =  ? 

26.  68,541  mi.  -r-  341  7m.  48.  444,312  -^  825  =  ? 

49.  $821.52  ^  63  +  $7.19  x  8  =  ? 

50.  $5,099.22  ^  57  +  $3.74  x  156  =  ? 

51.  6,574  -^  346  +  12,325  -^  29  =  ? 

52.  17,612  ^  518  +  10,323  -^  279  =  ? 

53.  (4,320  +  6,003)  -^  (109  +  170)  =  ? 

54.  (63,481—20,900)  -^  (119  _  70)  =  ? 

55.  ($69.80  —  $12)  -H  ($1.15  +  $2.25)  =  ? 


58  SIMPLE     NUMBERS. 

CONTRACTIONS    IN    DIVISION. 

4:2.  The  operation  of  division  may  often  be  shortened 
when  the  divisor  can  be  separated  into  factors.  In  this 
case,  we  divide  the  dividend  by  one  of  the  factors ;  we  then 
divide  the  quotient  by  another  factor ;  and  so  on,  till  all 
the  factors  have  been  used  (Arts.  34,  37). 

EXAM  PLES. 

1.  Divide  19,866  by  77,  that  is,  by  11  x  7. 

OPERATION. 

Explanation. — Here  we  divide  the  dividend  ^|  \  iq^qq 

by  11,  and  then  we  divide  the  resulting  quotient  

by  7.     Because  each  division  is  exact  there  is  '  )  1806 

no  remainder.  258  Ans. 

2.  Divide  1,592  by  35,  that  is,  by  7  x  5. 

Explanation. — Here  we  di-         operation. 
vide  in  succession  by  7  and  by  5.      7  )  1592 
The  first  step  shows  that  1,592 

contains  227  sevens  and  3  ones;        ^  )  ^^^  •   ^     ^'""^^  remainder, 
the  second  step  shows  that  227  45   .  2     Second  remainder. 

se'vens  contains  45  thirty  jives  and 

2  sevens.  But  2  sevens  are  equal  to  14  ones;  hence,  the  true  remainder 
is  14  +  3  or  17,  and  the  result  of  the  operation  is  45^f .  Hence,  to  find 
the  true  remainder  in  all  similar  cases  we  multiply  the  second  remain- 
der hy  the  first  divisor,  and  to  the  result  add  the  first  remainder. 

In  like  manner  to  find  the  true  remainder  when  there  are  three 
partial  divisions,  we  multiply  the  third  remainder  hy  the  second  divi- 
sor, and  to  the  product  add  the  second  remainder,  we  then  multiply  tfiis 
result  hy  the  first  divisor,  and  to  the  product  add  the  first  remainder. 

3.  Divide  7,445  by  84,  that  is,  by  7  x  4  x  3. 

Explanation. — Here  the  several  divisors  are  7,  4,  and  3  ;  the 
corresponding  remainders  are  4,  3,  and  1  ;  and  the  final  quotient  is 
88.  The  true  remainder  is  (1  x  4  4-  3)  x  7  +  4,  or  53  ;  hence,  the 
result  of  the  division  is  88^ 4 . 

4.  8,154  ->  99  =  ?  5.  1,913,578  -^  42  =  ? 


DIVISION.  59 

6.  15,336  -^  72  =  ?     9.  1,461,870  -v-  7  x  5  x  3  =  ? 

7.  93,312  -^  108  =  ?    10.  26,964  -^  11  x  5  x  2  =  ? 

8.  674,201  -^  110  =  ?   11.  93,696  -^  11  x  7  x  3  =  ? 

To  divide  by  10,  100,  1000,  &c.,  we  point  off  as  many  figures  from 
the  right  of  the  dividend  as  there  are  ciphers  in  the  divisar  ;  the  part 
on  the  left  is  the  quotient  and  the  part  on  the  right  is  the  remainder ^ 
(Art.  30). 

12.  Divide  8,759  by  100.  An8,   SH^. 

13.  746  by  10.    15.  4,981  by  100.   17.  3,425  by  100. 

14.  1,382  by  100.  16.  8,637  by  1000.  18.  94,276  by  10. 

Note. — If  the  divisor  is  composed  of  a  significant  part  followed 
by  ciphers,  we  cut  off  the  ciphers  and  also  the  same  number  of  fig- 
ures from  the  right  of  the  dividend  ;  we  then  divide  the  remaining 
part  of  the  dividend  by  the  significant  part  of  the  divisor  ;  to  find 
the  true  remainder  we  annex  to  the  partial  remainder  the  figures 
cut  off  from  the  dividend. 

19.  Divide  37,843  by  2,500.  Ans.    16^%\. 
•Explanation, — The  operation  operation. 

is  equivalent  to  dividing  first  by  25  00  )  378  43  (  15 

100  and  then  by  25.      The  first  '                 ' 

partial  remainder  is  43,  the  sec-  

ond  partial  remainder  is  3,  and  128 

the  first  divisor  is  100  ;    hence,  ^ok 

by  the  rule,  we  have  the  true  

remainder  equal  to  3  x  100  +  43,  True  remainder,      343 

or  343. 

20.  Divide  98,742  by  1,700.  Ans.   68^^. 
Perform  tbe  following  indicated  divisions  : 

21.  8,436  -T-  2,100.  24.  2,564,310  -=-  84,000. 

22.  8,566  -^  2,500.  25.  217,896  -j-  7,200. 

23.  17,439  -^  1,700.  26.  1,310,741  -^  64,000. 

Note. — If  both  dividend  and  divisor  terminate  in  ciphers,  we 
strike  off  from  the  right  of  each  as  many  as  are  common  to  both,  and 
then  perform  the  division. 


60  SIMPLE     NUMBERS. 

27.  Divide  875,000  by  2,500. 

OPERATION. 

Explanation. — Striking  off  two  ar  rvrv  \  o^v-rx  r.r.  /  n^^ 

.  1       »  T   .         .    1    ^  .  25,00  )  87o0  00  (  350 

ciphers  from  eacli  is  equivalent  to  ^ 

dividing  both   by   100,  which   ob-  7o 

viously  does  not  affect  the  result-  225 

ing  quotient.  -j^p. 

Perform  the  following  divisions  : 

28.  1,831,200  by  240.  35.  98,710  by  8,400. 

29.  $1,350,500  by  3,G50.  36.  66,920  by  8,800. 

30.  087,500  yds.  by  27,500  yds.  37.  8,623,000  by  250. 

31.  201,600  by  3,600.  38.  47,890  by  8,600. 

32.  41,580  by  540.  39.  35,100  by  4,800. 

33.  71,820  by  87.  40.  $1,400  by  $270. 

34.  1,749,600  by  360.  41.  368,000  by  4,200. 

PRACTICAL    PROBLEMS. 

43.  The  following  problems  afford  exercises  in  review 
of  the  four  fundamental  operations,  Addition,  Subtrac- 
tion, Multiplication,  and  Division. 

1.  An  estate  worth  $41,185  was  divided  equally  among 
5  persons ;  what  was  the  share  of  each  ?      Aiis.   $8,237. 

2.  An  estate  worth  $41,185  was  divided  equally  among 
a  certain  number  of  heirs  so  that  each  received  $8,237 ; 
how  many  heirs  were  there  ?  Ans.  5. 

3.  The  capital  of  a  joint-stock  company  is  $13,125  and 
is  divided  into  175  shares ;  what  is  the  value  of  each 
share  ?  Ans.   $75. 

4.  If  a  ship  sails  5,712  miles  in  48  days,  how  many 
miles  does  she  sail  per  day  ?  Ans.   119. 

5.  If  a  ship  sails  114  miles  in  1  day,  how  many  days  will 
ittakeher  to  sail  2,622  miles  ?  Ans.   23. 


DIVISION.  61 

6.  A  farmer  paid  $13,216  for  a  farm  of  112  acres;  how 
much  did  he  pay  per  acre  ? 

7.  IIow  many  acres  of  land  can  be  bought  with  $26,432, 
at  the  rate  of  $59  per  acre  ? 

8.  In  a  field  of  corn  there  are  21,033  hills  and  each  row 
contains  171  hills;  how  many  rows  are  there  ? 

9.  The  mean  diameter  of  the  earth  is  7,912  miles,  and 
that  of  the  sun  is  854,496  miles ;  how  many  diameters  of 
the  earth  are  there  in  the  sun's  diameter  ? 

10.  A  grocer  bought  55,664  pounds  of  flour  put  up  in 
barrels,  each  of  which  contained  196  pounds ;  how  many 
barrels  were  there  in  the  lot  ? 

11.  There  are  4,032  yards  of  cloth  in  96  equal  pieces ; 
how  many  yards  are  there  in  each  piece  ? 

12.  A  field  produces  3,404  bushels  of  oats  at  the  rate  of 
37  bushels  per  acre ;  how  many  acres  are  there  in  the 
field? 

13.  Twenty  pieces  of  cloth  contain  39  yards  each  ;  32 
pieces  coutain  38  yards  each ;  and  17  pieces  contain  43 
yards  each  ;  how  many  yards  in  all  ? 

14>  A  merchant  bought  1 75  yards  of  cloth  at  7  dollars 
per  yard  and  afterwards  sold  72  yards  at  9  dollars  per  yard 
and  the  remainder  at  8  dollars  per  yard ;  how  much  did 
he  gain  ? 

15.  A  dealer  bought  27  barrels  of  flour  at  $14  per  barrel 
and  gave  in  exchange  32  cords  of  wood  at  $8  per  cord  and 
paid  the  balance  in  cash ;  how  much  cash  did  he  pay  ? 

16.  A  man's  income  is  $3,150  per  year  and  his  expenses 
are  $2,817  per  year ;  how  much  can  he  save  in  6  years  ? 

17.  A  farmer  bought  32  acres  of  land  at  $95  per  acre, 


6'i  SIMPLE     NUMBEKS. 

71  acres  at  $47  per  acre,  38  acres  at  $62  per  acre,  and  19 
acres  at  $88  per  acre  ;  what  did  he  pay  for  the  whole  ? 

18.  The  factors  of  one  number  are  19,  17,  and  23;  of 
ano-ther  number,  31,  29,  and  11 ;  and  of  a  third  number, 
77  and  83  :  what  is  the  sum  of  the  numbers  ? 

19.  Two  men  set  out  from  the  same  point  and  travel  in 
opposite  directions;  the  first  travels  at  the  rate  of  43 
miles  per  day,  and  the  second  at  the  rate  of  37  miles  per 
day :  how  far  apart  are  they  at  the  end  of  7  days  ? 

20.  A  farmer  bought  6  oxen  at  $65  each,  12  cows  at  $42 
each,  and  142  sheep  at  16  each  ;  what  did  he  pay  for  the 
whole  ? 

21.  In  a  freight  car  there  are  6  boxes  of  goods,  each 
weighing  382  pounds;  13  barrels,  each  weighing  218 
pounds ;  and  37  bags,  each  weighing  179  pounds :  how 
many  pounds  in  all  ? 

22.  In  19  bales  of  cloth,  each  bale  containing  16  pieces, 
and  each  piece  containing  42  yards,  how  many  yards  ? 

23.  What  number  multiplied  by  86  will  give  the  same 
product  as  163  multiplied  by  430  ? 

24.  How  many  yards  of  muslin  at  14  cents  a  yard  must 
be  given  in  exchange  for  35  bushels  of  oats  at  56  cents  a 
bushel  ? 

25.  A.,  B.,  and  0.  enter  into  partnership;  A.  puts  in 
$7,200,  B.  puts  in  1700  more  than  A.,  and  C.  puts  in  $550 
less  than  A.  and  B.  together :  what  is  the  capital  of  the 
firm? 

26.  A.'s  income  is  5  times  B.'s,  B.'s  income  is  3  times 
C.'s,  and  C.'s  income  is  $1,325  ;  what  is  the  entire  income 
of  A.,  B.,  and  0,  '^ 


DlVISIOl^.  63 

27.  A  farmer  bought  154  acres  of  land  at  $64  per  acre, 
and  sold  the  whole  for  111,704 ;  how  many  dollars  did  he 
gain  per  acre  ? 

28.  The  distance  from  New  York  to  Albany  is  144 
miles,  and  each  mile  contains  5,280  feet ;  how  many  hours 
will  it  take  a  man  to  walk  from  New  York  to  Albany  if  he 
walks  at  the  rate  of  352  x  60  feet  an  hour  ? 

29.  The  sum  of  two  numbers  is  10,370,  and  the  second 
is  4  times  the  first ;  what  are  the  numbers  ?  t 

30.  The  first  of  three  numbers  is  24,  the  second  is  3 
times  the  first,  and  the  third  is  4  times  the  sum  of  the  first 
and  second ;  what  is  the  difference  between  the  second 
and  third  ? 

31.  Write  down  4,617,  multiply  it  by  12,  divide  the  pro- 
duct by  9,  add  365  to  the  quotient,  and  from  the  sum  sub- 
tract 5,521 ;  what  is  the  final  result  ? 

32.  Mrs.  White  has  3  houses  worth  $12,530,  $11,324, 
and  ^9,875,  also  a  farm  worth  $6,720.  To  her  daughter 
she  gave  one  third  the  value  of  the  houses  and  one  fourth 
the  value  of  the  farm,  and  then  she  divided  the  remainder 
equally  between  her  two  sons;  how  much  did  each  re- 
ceive ? 

*  33.  What  IS  the  difference  between  the  cost  of  425  sheep 
at  $4.75  apiece  and  38  cows  at  $48.25  apiece  ? 

34.  The  distance  from  Chicago  to  San  Francisco  is  2,448 
miles  ;  how  long  will  it  take  a  man  to  walk  the  whole  dis- 
tanca  at  the  rate  of  24  miles  a  day  ? 

35.  Two  men  had  an  equal  interest  in  a  herd  of  cattle ; 
one  took  72  at  $35  apiece  and  the  other  took  the  rest  at 
$42  apiece ;  how  many  cattle  in  the  herd  ? 


64  SIMPLE     NUMBERS. 

36.  A  man  bought  4  horses  at  $116  apiece  and  2  colts 
at  156  apiece,  and  paid  for  them  in  flour  at  $12  a  barrel ; 
how  many  barrels  of  flour  did  it  require  to  make  the  pay- 
ment ? 

37.  A  man  travels  due  north  for  7  days  at  the  rate  of 
37  miles  a  day ;  he  then  returns  on  his  path  at  the  rate  of 
29  miles  a  day;  how  far  is  he  from  the  starting  point  at 
the  end  of  12  days  trayel  ? 

^.  A  man  bought  742  acres  of  land  at  118  an  acre  ;  he 
sold  at  one  time  211  acres  at  $22  an  acre  and  at  another 
time  he  sold  184  acres  at  125  an  acre ;  at  what  rate  per 
acre  must  he  sell  the  rest  to  gain  13,867  ? 

39.  A.  bought  a  farm  for  $3,612 ;  he  sold  half  of  it  at 
156  an  acre  and  received  for  it  $2,408 :  how  many  acres 
did  he  buy  and  what  did  he  give  per  acre  ? 

40.  How  many  horses  worth  $112  apiece  can  be  bought 
for  28  oxen  woith  $63  each,  52  cows  worth  $42  each,  175 
sheep  worth  $6  each,  and  $2,394  in  cash  ? 

REVIE^A;'      QUESTIONS. 

(35.)  What  is  division?  Define  the  dividend  ;  the  divisor;  the 
quotient  ;  and  the  remainder.  When  is  division  exact  ?  What  are 
factors  ?  (36.)  What  does  the  sign  of  division  indicate  when  placed 
between  two  numbers  ?  In  what  other  ways  may  division  be  indi- 
cated ?  (37.)  What  is  the  relation  between  multiplication  and  divi- 
sion? (38.)  State  the  leading  principle  of  division.  (30.)  What 
is  short  division  ?  What  is  long  division  ?  (40.)  Give  the  rule  for 
short  division.  Method  of  Proof?  (41.)  Give  the  rule  for  long 
division.  (42.)  How  do  you  divide  by  a  composite  number  ?  What 
is  the  method  of  determining  the  true  remainder  ?  How  do  you 
divide  by  10,  100,  1,000,  etc.  ?  How  do  you  divide  by  a  number  that 
ends  in  ciphers  ?  How  do  you  divide  when  both  dividend  and  divi- 
sor terminate  in  ciphers  ?  (43.)  What  are  the  fundamental  opera- 
tions of  arithmetic  ? 


FACTORING     AI^D     CANCELLIKG.  65 

V.    FACTORING   AND   CANCELLING. 

DEFINITIONS. 

44.  A  Factor  of  a  number  is  one  of  its  exact  integral 
divisors,  (Art.  35).     Thus,  2,  3,  and  4,  are  factors  of  12. 

45.  A  Composite  number  is  a  number  composed  of 
two  or  more  Integral  Factors  (Art.  34).  Thus,  15  is  a 
composite  number,  because  it  is  the  product  of  3  and  5. 

A  Prime  number  is  one  that  cannot  be  separated  into 
any  integral  factors  except  1  and  the  number  itself.  Thus, 
2,  3,  5,  etc.,  are  prime  numbers. 

46.  Factoring  is  the  operation  of  separating  a  num- 
ber into  integral  factors. 

The  factors  of  a  number  may  be  either  prime;  or  composite.  Com- 
posite factors  may  themselves  be  factored,  and  so  on,  till  all  the  fac- 
tors are  prime.  Thus,  24=2  x  12=2  x  2  x  6=2  x  2  x  2  x  3 ;  hence,  the 
prime  factors  of  24  are  2,  2,  2,  and  3. 

MENTAL     EXERCISES. 

1.  What  is  the  product  of  2  and  3  ?  What  are  the  fac- 
tors of  6  ?  of  9  ?  of  15  ?  of  77  ?  of  121  ?  of  144  ? 

2.  What  is  the  continued  product  of  2,  3,  and  7  ?  of 
2  and  3  ?  of  2  and  7  ?  of  3  and  7  ?  What  are  the  prime 
factors  of  42  ?  What  are  the  composite  factors  of  42  ? 
In  how  many  ways  may  42  be  factored  ? 

Note. — Because  every  number  is  the  product  of  1  and  of  the 
number  itself,  these  numbers  are  not  specially  considered  in  the 
operation  of  factoring, 

3.  What  are  the  prime  factors  of  4  ?  of  9  ?  of  27  ?   of 

81  ?  of  121  ?  of  143  ?  of  187  ?  of  198  ?  of  225  ? 

Note. — The  product  of  two  or  more  factors  that  are  equal  is 
called  a  power.    The  name  of  the  power  depends  on  the  number  of 
8 


66  SIMPLE     KUMBEE8. 

equal  factors.  Thus,  3  x  3,  or  9,  is  the  second  power,  or  the  squa/re 
of  3  ;  3  X  3  X  3,  or  27,  is  the  third  power,  or  the  cube  of  3;  3x3x3x3, 
or  81,  is  ihe  fourth  potcer  of  3  ;  and  so  on.  Every  lower  power  of  a 
number  is  a  factor  of  a  higher  power  of  the  same  number. 

4.  What  are  the  factors  of  81  ?  of  27  ?  of  9  ?  How 
many  prime  factors  has  81  ?  How  many  has  27  ?  How 
many  has  9  ?  What  is  the  square  of  6  ?  the  third  power 
of  5  ?  the  fourth  power  of  4  ?  the  fifth  power  of  2  ? 

5.  What  is  the  second  power  of  10  ?  the  third  power  of 
10  ?  the  fourth  power  of  10  ?  How  many  ciphers  are 
required  to  write  the  square  of  10  ?  the  cube  of  10  ? 

6.  What  is  the  fifth  power  of  10  ?  How  many  ciphers 
in  the  fifth  power  of  10  ?  What  power  of  10  is  100  ? 
1,000  ?  10,000  ?  100,000  ?  1,000,000  ? 

PRINCIPLES    OF    FACTORING. 

47.  The  operation  of  resolving  a  number  into  prime 
factors  depends  on  the  following  principles : 

1°.  A  nicmher  is  equal  to  the  continued  product  of  all  its 
prirne  factors. 

2°.  If  a  number  is  divided  hy  one  of  its  prime  factors, 
the  quotient  is  equal  to  the  continued  product  of  all  the 
others. 

OPERATION    OF    FACTORING. 

48.  Let  210  be  separated  into  prime  factors. 
Explanation.— We  first  divide  by  2,  which  is  a        operation. 

prime  factor  ;  we  next  divide  the  first  quotient  by  2)210 

3,  which  is  also  a  prime  factor  ;  we  then  divide  the  3)105 

second  quotient  by  5,  and  find  7  for  a  quotient.  — — 

The  numbers  2,  3,  5,  and  7  are  the  required  factors ;  ^)j^ 

that  is,  7 

210  =  2  X  3  X  4  X  5. 


FACTOEIN"G     AKD     CAKCELLIKG.  67 

In  like  manner  other  composite  numbers  may  be  factored  ;  hence, 
the  following 

RULE. 

Divide  the  given  nurriber  by  one   of  its   prime 

factors;    then  divide    the    quotient   by  one  of  its 

prime  factors ;    and  so  on,  till  a  quotient  is  found 

that  is  a  prime  number ;  the  several  divisors  and 

the  last  quotient  are  the  required  factors. 

Note. — It  will  be  found  convenient  to  begin  the  division  with 
the  smallest  prime  factor. 

EXAMPLES. 

Resolve  the  following  numbers  into  their  prime  factors : 

1.  42.  Ans.  2x3x7. 

Note. — If  there  are  more  than  two  factors  in  any  indicated  prod- 
uct, the  sign  of  multiplication  may  be  replaced  by  a  simple  dot ; 
thus,  3  .  3  .  7  is  equivalent  to  2  x  3  x  7. 

2.  180.   Ans.  2.2.3.3.5.    5.  770.  Ans.  2  .  5  .  7  .  11. 
3.378.  ^ws.  2.3.3.3.7.    6.  1,575.  ^w.<?.  3  .  3  .  5.5.7. 
4.  330.  Ans.  2.3.5.11.      7.  3,850.  Ans,  2 .  5  .  5 .  7 .  11. 
The  operation  of  factoring  is  principally  performed  by 

inspection  and  trial.    It  may  sometimes  be  facihtated  by 
using  the  following 

TABLE    OF    PRIME    NUMBERS    FROM    I    TO    150. 

1        2        3        5        7      11       13      17      19 

23      29      31      37      41      43      47      53      59 

61      67      71      73      79      83      89      97    101 

103     107     109    113     127     131     137     139     149 

Find  the  prime  factors  of  the  following  numbers : 

8.  402.  10.  3,290.  12.   1,095. 

9.  1,659.  11.  1,554.  13.  2,310. 


68  SIMPLE     KUMBEES. 

14.  2,730.  16.  7,644.  18.      786. 

15.  17,160.  17.  1,872.  19.  3,136. 

Note.— If  the  final  digit  of  a  number  is  0,  2,  4,  6,  or  8,  the  num 
ber  is  divisible  by  2. 

If  the  sum  of  the  digits  of  a  number  is  divisible  by  3,  the  num- 
ber itself  is  divisible  by  3. 

If  the  final  digit  of  a  number  is  5,  the  number  is  divisible  by  5. 

20.  930.  23.  1,738.  26.  1,105. 

21.  1,455.  24.  3,255.  27.  3,171. 

22.  3,685.  25.  1,001.  28.  2,873. 

CAN  CELLATI ON. 

49.  Cancellation  is  the  operation  of  striking  out  one 
or  more  factors  that  are  common  both  to  the  dividend  and 
the  divisor  of  an  indicated  division. 

The  operation  is  performed  by  drawing  a  line  across  the 

factor  that  is  to  be  struck  out,  or  cancelled.    Thus,  in  the 

^.  $.4 
expression       ^       the  factors  2  and  3  are  cancelled. 

OBJECT    AND    PRINCIPLES    OF    CANCELLATION. 

50,  The  operation  of  division  may  sometimes  be  short- 
ened by  cancelling  factors  common  to  both  dividend  and 
divisor.     This  method  depends  on   the  following  prin- 


1°.  Striking  out  a  factor  of  a  number  is  equivalent  to 
dividing  the  number  by  that  factor. 

2°.  If  both  dividend  and  divisor  are  divided  by  the  same 
number  the  quotient  is  not  changed. 

MENTAL     EXERCISES. 

1.  What  is  the  quotient  of  3  x  8  by  3  x  4  ?  of  8  by  4  ? 
What  is  the  effect  of  cancelling  3  in  both  dividend  and 
divisor? 


FACTORING     AND     CANCELLING.  69 

2.  Divide  3  x  3  x  7  by  3  x  3  x  2.  7  by  2.  7x7 
by  7  X  2. 

A  PP  LI  CATI  O  N  S. 

51.  Let  it  be  required  to  divide  2  .  5  .  7 .  11  by  5  .  7 .  7. 

Explanation. — Hav-  operation. 

ing  indicated  the  divi-     2.5.  7^  _  2.^.^.11  _  22  _ 
sion,  we   cancel  all  the  W7Y.~7  WTfT^  "t" 

factors  common  to  the 
dividend  and  divisor  ;  we  then  divide  2  x  11,  or  22,  by  7. 

In  like  manner  we  treat  all  similar  cases  ;  hence,  the 

RULE. 

/.  Indicate  the  division,  and  strihe  out  all  the 
factors  that  are  common  to  the  dividend  and 
divisor. 

II.  Divide  the  product  of  the  factors  that  remain 
in  the  dividend  by  the  product  of  those  that  remain 
in  the  divisor. 

EXAMPLES. 

Perform  the  following  operations,  using  the  method  of 
cancellation : 

1.  2 .  3  .  5  .  7  .  11  -^  2  .  3  .  5  .  19.  Ans.  fj  =  4^^. 

2.  3. 3. 3. 5. 5. 7.  13 -^2. 2.  2.  3. 3. 3. 5.  5. 

Ans.   11  f. 

Note.  —If  all  the  factors  in  either  dividend  or  divisor  are  can 
celled,  the  unit  1  will  be  left  ;  if  it  is  in  the  dividend  it  must  be 
retained,  if  it  is  in  the  divisor  it  may  be  omitted. 

4.  2.3.5.7-^2.2.5.7.8.  Ans.   ^. 

5.  3  .  5  .  5  .  7  .  13  -^  3  .  5  .  7.  Ans.   65. 

Note. — The  operation  of  cancellation  should  be  performed  men- 
tally. 

6.  Divide  144  x  56  by  96.  Ans.   84. 


70  SIMPLE     NUMBERS. 

Explanation.— Having  indi-  operation. 

cated  the  division,  we  see  that  12      •^'^  * 

is  a  factor  of  144  and  of  96 ;  we     ^^^  X  56  _  12  X  $^  _  g^ 
therefore  mark  out  144,  replacing  00  $ 

it  by  13,  and  96,  replacing  it  by  8,  8 

as  shown  in  the  margin.     This 

reduces  the  operation  to  dividing  12  x  56  by  8.  We  now  see  that  8 
is  a  factor  of  56  and  also  of  8  ;  we  therefore  cancel  56,  replacing  it 
by  7,  and  8,  dropping  the  factor  1.     The  result  is  84. 

Perform  the  following  iudicafced  operations : 

7.  168  X  216  ^  42  X  54.    16.  97  x  30  -^  37  x  3. 

8.  9  X  24  X  31  -^  72.  17.  161  x  15  -^  161  x  3. 

9.  364  X  42  ^  14  X  21.      18.  17  x  11  x  4  -^  34  x  22. 

10.  36  X  37  -^  27  x  11.  19.  26  x  33  -r-  46  x  11. 

11.  48  X  125  -^  3  X  5.  20.  34  x  26  -^  39  x  17. 

12.  342  X  6  -^  36.  21.  57  x  18  x  4  -=-  34  x  8. 

13.  1,323  X  5  -^  63  X  5.  22.  114  x  22  -v-  11  x  76. 

14.  147  X  9  X  5  -^  22  x  21.  23.  170  x  55  -^  85  x  55. 

15.  27  X  200  -^  18  X  56.  24.  169  x  7  -^  13  x  14. 

PRACTICAL      PROBLEMS. 

1.  How  many  boxes  of  tea,  each  containing  24  pounds, 
at  75  cents  a  pound,  must  be  given  for  145  bags  of  wheat, 
each  bag  containing  2  bushels,  at  180  cents  a  bushel  ? 

Ans.  29  boxes. 
Solution.    145  x  2  x  180  -f-  24  x  75  =  29  boxes,  Ans. 

2.  A.  worked  18  days  at  13  per  day,  for  which  he  re- 
ceived 6  barrels  of  flour ;  how  much  was  the  flour  worth 
per  barrel  ?  Ans.  19. 

3.  A  man  buys  3  pieces  of  cotton  cloth,  each  containing 
42  yards,  at  13  cents  per  yard,  and  pays  for  it  in  butter  at 
21  cents  per  pound  ;  how  many  pounds  of  butter  must  he 
give  ?  Ans,  78  lbs. 


FACTORIIfG     AND     CANCELLIN^G.  71 

4.  Bought  15  barrels  of  apples,  each  containing  3  bush- 
els, at  84  cents  a  bushel ;  how  many  cheeses,  each  weigh- 
ing 45  pounds,  at  12  cents  per  pound,  will  pay  for  the 
apples  ?  Ans.  7  cheeses. 

5.  A  farmer  sold  2  loads  of  hay  each  weighing  2,240  lis., 
at  1  cent  a  pound,  for  which  he  received  4  pieces  of  sheet- 
ing, each  containing  40  yards  j  what  did  the  sheeting  cost 
per  yard  ? 

6.  How  many  bushels  of  corn  at  93  cfs.  a  bushel  will 
pay  for  2  barrels  of  sugar  each  barrel  containing  372  lbs., 
the  sugar  being  worth  8  cfs.  a  pound  ? 

7.  A  man  received  for  9  days'  work,  3  barrels  of  flour 
worth  $6  a  barrel;  what  did  he  receive  for  each  day's 
work  ? 

8.  How  many  firkins  of  butter,  each  containing  50  lis., 
at  18  cts.  a  pound,  must  be  given  for  3  bar.  of  sugar,  each 
containing  200  lbs.,  at  9  cts.  a  pound  ? 

9.  How  many  boxes  of  tea,  each   containing  24  lbs., 
worth  75  cts.  a  pound,  must  be  given  for  4  bins  of  wheat,' 
each  containing  145  bushels,  and  worth  11.80  a  bushel  ? 

10.  How  many  pounds  of  butter  at  24  cts.  a  pound  will 
buy  18  yds.  of  cotton  at  36  cts.  a  yard  ? 

REVIEV7  QUESTIONS. 
(44.)  What  is  a  factor  of  a  number  ?  (45.)  What  is  a  composite 
number  ?  A  prime  number  ?  Illustrate.  (46.)  What  is  factoring  ? 
(47.)  What  are  the  principles  of  factoring?  (48.)  What  is  the 
rule  for  resolving  a  number  into  prime  factors  ?  When  there  are 
more  than  two  factors  in  a  product,  how  may  the  multiplication  be 
indicated  ?  (49.)  What  is  cancellation  ?  How  is  it  performed  ? 
(50.)  What  is  the  object  of  cancellation  ?  On  what  principles  does 
it  depend?  (51.)  GUve  the  rule  for  shortening  division  by  cancel- 
lation. 


72  SIMPLE     NUMBERS. 


VI.  GREATEST  COMMON  DIVISOR 
AND  LEAST  COMMON  MULTI- 
PLE. 

DEFINITIONS. 

52.  A  Common  Divisor  of  two  or  more  numbers  is 

a  number  that  will  exactly  divide  each  of  them.     Thus,  4 

is  a  common  divisor  of  8,  16,  and  32. 

A  common  divisor  of  two  or  more  numbers  is  also  called  a  com- 
mon measure  of  those  number. 

The  Greatest  Common  Divisor  of  two  or  more  num- 
bers is  the  greatest  number  that  will  exactly  divide  them- 
all.  Thus,  8  is  the  greatest  common  divisor  of  8,  16, 
and  32. 

Numbers  that  have  no  common  divisor,  except  1,  are 
said  to  be  prime  with  respect  to  each  other. 

MEISTTAL      EXERCISES. 

1.  What  number  will  exactly  divide  both  15  and  20  ? 
What  is  their  common  divisor  ? 

2.  Name  all  the  exact  divisions  of  30 ;  of  42.  What 
divisors  are  common  to  30  and  42  ?  What  is  their  greatest 
common  divisor  ? 

3.  What  are  the  prime  factors  of  70  ?  of  50  ?  What 
prime  factors  are  common  to  70  and  50  ?  What  is  their 
product  ?  Wliat  is  the  greatest  common  divisor  of  70 
and  50  ? 

4.  What  are  the  prime  factors  of  18  ?  of  12  ?  of  18—12  ? 
What  is  the  greatest  common  divisor  of  18  and  12  ?  of 
18  and  18  —  12  ?  of  18  —  12  and  8  ?  What  is  the  greatest 
common  divisor  of  121  and  99  ? 


GREATEST     COMMON     DIVISOR.  73 

METHODS     AND      PRINCIPLES. 

53.  There  are  two  methods  of  finding  the  greatest  com- 
mon divisor  of  two  or  more  numbers :  1°.  By  factors  ; 
and  2°.  By  continued  division. 

Both  methods  depend  on  the  following  principles : 
1°.  Any  factor  common  to  two  or  more  numbers  is  a  com- 
mon divisor  of  those  numbers. 

2°.  The  greatest  co7nmon  divisor  is  equal  to  the  continued 
product  of  the  -prime  factors  that  are  common  to  all  the 
numbers. 

METHOD      BY      FACTORS. 

54.  Let  it  be  required  to  find  the  greatest  common 
divisor  of  126,  210,  and  546. 

Resolving  the  numbers  into  prime  factors,  we  have, 
126=2.3.3.7,  210  =  2.3.5.7,   and   546  =  2.3.7.13. 

The  factors  2,  3,  and  7  are  common  to  all  the  numbers, 
and  they  are  the  only  ones  that  are  common ;  hence,  their 
product  is  the  greatest  common  divisor  of  the  given  num- 
bers ;  denoting  the  greatest  common  divisor  by  the  initials 
g.  c.  d.,  we  have  g.  c.  d.  =  42. 

All  similar  cases  may  be  treated  in  like  manner ;  hence, 
the  following 

RULE. 

/.  Resolve  the  numbers  into  prime  factors. 
II.  Find  the  cojitinued  product  of  the  prime  fac- 
tors common  to  all  the  numbers. 

EXAMPLES. 

1.  What  is  th3  greatest  common  divisor  of  168,  216, 
and  408,  that  is,  of  2.2.2.3.7,  2.2.2.3.9,  and 
2.2.2.3.17?  Ans.  2  .  2  .  2  .  3  =  24. 


74 


SIMPLE     NUMBEES. 


Find  the  greatest  common 
groups  of  numbers : 

2.  408,  and  740. 

3.  90,  315,  and  810. 

4.  441,  and  567. 

5.  195,  285,  and  315. 

6.  462,  726,  and  1,254. 

7.  1,470,  2,310,  and  2,730. 

8.  320,  1,216,  and  6,400. 


divisors  of  the  following 

9.  540,  648,  and  756. 

10.  567,  648,  and  729. 

11.  84,  126,  and  210. 

12.  4,410,  and  3,150. 

13.  1,335,  and  1,869. 

14.  1,584,  and  1,188. 

15.  26,195,  and  273. 


OPEEATION. 

Dividend. 


Quotient. 


ADDITIONAL     PRINCIPLE. 

55,  The  greatest  common  divisor  of  two  numbers  will 
divide  their  remainder  after  division. 

For,  let  8,  which  is  the  greatest  common  divisor  of  88 
and  24,  be  taken  as  a  unit :  the  two  numbers,  expressed 
in  ferms  of  this  unit,  are  11  eights  and  3  eights.  Let  the 
greater  number  be  divided  by  the  less. 

Explanation.  —  We 

see  that  11  eights  contains 
3  eights,  3  times  with  a 
remainder  equal  to  3 
eights. 

Now,  11  eights,  3  eights, 
and  S  eights  are  all  divis- 
ible by  8,  and  the  quotients  are  prime  with  respect  to  each  other  ; 
hence,  8  is  the  greatest  common  divisor  of  3  eights  and  2  eights,  as 
well  as  of  11  eights  and  3  eights,  that  is,  the  greatest  common  divisor 
of  the  given  numbers  is  also  the  greatest  common  divisor  of  the  less 
number  and  of  their  remainder  after  division  :  hence,  the  following 

PRINCIPLE. 

The  greatest  common  divisor  of  two  numbers  is  the  same 
as  the  greatest  common  divisor  of  the  smaller  number  and 
of  their  remainder  after  division. 


Divisor. 

3  eights)ll  eights(3 
9  eights 


2  eights  .  .  .  Remainder. 


GREATEST     COMMON     DIVISOR.  75 

METHOD     BY     CONTINUED     DIVISION. 

56.  Let  it  be  required  to  find  the  greatest  conimon 
divisor  of  88  and  24 

Explanation. — Dividing  88  by  24,  we  operation. 

find  16  for  a  remainder  ;    then  dividing  24)88(3 

24  by  16,  we  find  8  for  a  remainder  ;  then  72 

dividing  16  by  8,  we  find  0  for  a  remain-  16^24n 

der.     Hence,  8  is  the  greatest  common  16 

divisor  of  16  and  24,  (Art.   55) ;  it  is  — 

therefore  the  greatest  common  divisor  of  8)16(2 

24  and  88.  16 

In  like  manner,  all  similar  cases  may  a 

be  treated  ;  hence,  the  following 

RULE. 
J.  Divide  the  greater  number  by  the  less  and  find 
the  remainder. 

II.  Take  the  divisor  for  a  new  dividend  and  the 
remainder  for  a  divisor,  and  proceed  as  before. 

III.  Continue  the  operation  till  a  remainder  is 
fou7%d  that  will  exactly  divide  the  preceding  divi- 
sor ;  this  will  be  the  greatest  com^mon  divisor  of  the 
given  numbers. 

EXAMPLES. 

Find  the  greatest  common  divisors  of  the  following 
groups  of  numbers : 

1.  3,471  and  2,13G.  8.  10,353  and  14,877. 

2.  1,584  and  3,168.  9.  4,410  and  5,670. 

3.  2,898  and  7,866.  10.  3,471  and  1,869. 

4.  3,724  and  5,852.  11.  1,584  and  2,772. 

5.  3,444  and  2,268.  12.  432  and  1,224. 

6.  10,395  and  16,797.  13.  945  and  3,240. 

7.  667  and  391.  14.  1,080  and  1,224. 


76  SIMPLE     NUMBERS. 

To  find  tlie  greatest  common  divisor  of  more  than  two  n  ambers, 
begin  with  the  least  and  find  tlie  greatest  common  divisor  of  two, 
then  of  that  result  and  the  third,  then  of  that  and  the  fourth,  and 
so  on  ta  the  last. 

15.  805,  1,311,  and  1,978.  21.  740,  999,  and  1,147. 

10.  504,  5,292,  and  1,512.  22.  108,  216,  and  324. 

17.  837,  1,134,  and  1,347.  23.  803,  949,  and  1,241. 

18.  492,  744,  and  1,044.  24.  836,  988,  and  1,444. 

19.  944,  1,488,  and  2,088.  25.  935,  1,045,  and  1,265. 

20.  216,  408,  and  740.  26.  581,  1,079,  and  913. 

LEAST    COMMON     MULTIPLE. 

DEFINITIONS. 

57.  A  Multiple  of  a  number  is  a  number  that  is 
exactly  divisible  by  it.     Thus,  12  is  a  multiple  of  6. 

A  Common  Multiple  of  two  or  more  numbers  is  a 
number  that  is  exactly  divisible  by  each.  Thus,  48  is  a 
common  .multiple  of  4,  6,  and  8. 

The  Least  Common  Multiple  of  two  or  more  num- 
bers is  the  least  number  that  is  exactly  divisible  by 
each.  Thus,  24  is  the  least  common  multiple  of  4,  6, 
and  8. 

MENTAL      EXERCISES. 

1.  What  are  the  prime  factors  of  4  ?  of  6  ?  of  12  ? 
What  is  the  least  number  that  can  be  divided  by  both 
4  and  6  ?  What  is  the  least  common  multiple  of  4 
and  6  ?  of  12  and  30  ?  of  18  and  48  ? 

2.  How  many  times  is  2  a  factor  of  18  ?  of  60  ?  of  180  ? 
How  many  times  is  3  a  factor  of  18  ?  of  60  ?  of  180  ? 
Are  there  any  prime  factors  in  18,  or  in  60,  not  in  180? 
What  is  the  least  common  multiple  of  18  and  60  ? 


LEAST     COMMOif     MULTIPLE.  77 

PRINCIPLES. 

58.  The  operation  of  finding  the  least  common  multi- 
ple of  two  or  more  numbers  depends  on  the  following 
principles : 

1°.  The  least  common  multiple  must  contain  every  prime 
factor  of  each  number, 

2°.  It  must  contain  every  prime  factor  the  greatest  num- 
ber of  times  it  enters  any  of  the  numbers. 

OPERATION    OF    FINDING    THE    LEAST    COMMON    MULTIPLE. 

59.  Let  it  be  required  to  find  the  least  common  multi- 
ple of  12,  25,  and  90. 

Resolving  the  given  numbers  into  prime  factors,  we  have, 

12  =  2.2.3,     25  =  5.5,     and    90  =  2.3.3.5. 

The  least  common  multiple  must  contain  the  factor  2 
twice  to  be  divisible  by  12,  it  must  contain  the  factor 
3  twice  to  be  divisible  by  90,  and  it  must  contain  the 
factor  5  twice  to  be  divisible  by  25.  Hence,  if  we  de- 
note the  least  common  multiple  by  the  initials  /.  c.  m,, 

we  have, 

Z.  c.  ?7i.  =  2  .  2  .  3  .  3  . 5  .  5  =  900. 

The  practical  method  of  factoring  is  as  follows: 

Explanation. — Having  written  the  num-  operation. 

bers  in  a  line,  we  see  by  inspection  that  2  is 
a  prime  factor  of  12  and  90.  We  therefore 
write  2  on  the  left  as  a  divisor.  Dividing  13 
and  90  by  2,  we  write  the  quotients,  and  also 
the  undivided  number,  25,  in  the  second 
line.     We  then  see  that  5  is  a  prime  factor 

of  25  and  45  ;  writing  it  on  the  left  and  proceeding  as  before,  we 
find  the  numbers  in  the  third  line.  We  then  see  that  3  is  a  prime 
factor  of  6. and  9  ;  proceeding  as  before,  we  find  the  numbers  in  the 


2 

12,  25,  90 

5 

6,  25,  45 

3 

6,     5,     9 

2,     5,     3 

78  SIMPLE     NUMBERS. 

fourth  line,  all  of  which  are  prime  with  respect  to  each  other.  Here, 
we  have  resolved  12  into  the  factors  2,  3,  and  2  ;  25  into  the  factors 
5,  and  5  ;  and  90  into  the  factors  2,  5,  3,  and  3.  Hence,  from  the 
principles  of  Art.  58,  we  have  I.  c.  m.  =  2  .2  .3  .'S  .5  .  5  =  900. 

In  like  manner  we  can  find  the  least  common  multiple  of  any 
other  group  of  numbers  ;  hence,  the  following 

RULE. 

/.  Write  the  numbers  in  a  line  and  divide  by 
any  prime  factor  that  is  contained  in  two  or  more, 
writing  the  quotients  and  the  undivided  numbers 
in  the  line  below. 

II.  Then  operate  on  the  second  line  of  numbers 
in  the  same  manner,  and  so  on,  till  a  line  of  num- 
bers is  found  that  are  prime  with  respect  to  each 
other. 

III.  Find  the  continued  product  of  the  numbers 
in  the  last  line  and  of  the  divisors  used ;  this 
will  be  the  least  common  multiple  of  the  given 
numbers. 

EXAM  PLE  s. 

Find  the  least  common  multiple  of  the  following  groups : 

1.  3,  4,  8,  and  12.  11.  84,  100,  and  224. 

2.  6,  7,  8,  9,  and  10.  12.  49,  56,  63,  and  84. 

3.  4,  6,  9,  14,  and  16.  13.  20,  126,  150,  and  490. 

4.  12,  48,  18,  and  70.  14.  84,  150,  and  1,225. 

5.  14,  20,  198,  and  210.  15.  39,  52,  78,  and  117. 

6.  8,  18,  20,  and  70.  16.  130,  390,  and  338. 

7.  9,  18,  27,  36,  54,  45.  17.  136,  170,  and  425. 

8.  7,  15,  21,  28,  and  35.  18.  171,  285,  and  475. 

9.  15,  16, 18,  20,  and  24.  19.  275,  385,  and  539. 
10.  49, 14,  84, 168,  and  98.  20.  126, 189,  and  bQ. 


LEAST     COMMON     MULTIPLE.  79 

PRACTICAL      PROBLEMS. 

1.  What  is  the  g.  c.  d,  of  118  and  |45  ? 

%  What  is  the  I  c,  m.  of  I'ift,  and  90/^.  ? 

3.  A  farmer  has  225  hu.  of  oats,  135  hu.  of  wheat,  and 
90  bu.  of  rye,  which  he  wishes  to  put  in  bins  of  equal  size; 
each  bin  to  be  as  large  as  possible ;  how  many  bushels 
must  each  hold  that  all  may  be  filled  without  mixing  the 
different  kinds  of  grain  ? 

4.  What  is  the  shortest  piece  of  wire  that  can  be  cut  up 
into  exact  lengths  of  either  Qft.,  Sft,  or  10 ft.  ? 

5.  There  are  three  companies  of  soldiers  containing 
respectively  36,  60,  and  84  men,  each  of  which  is  to  be 
divided  into  platoons ;  how  many  men  must  be  put  in  a 
platoon,  so  that  all  the  platoons  shall  be  equal  and  each 
contain  the  greatest  possible  number  of  men  ? 

6.  How  many  quarts  are  there  in  the  smallest  cask  of 
cider  that  can  be  exactly  measured  by  either  a  3  quart, 
a  5  quart,  or  a  6  quart  measure  ? 

REVIEW     QUESTIONS. 

(52.)  What  is  a  common  divisor  of  two  or  more  numbers? 
What  is  the  greatest  common  divisor  of  two  or  more  numbers  ? 
When  are  numbers  prime  with  respect  to  each  other?  (53.) 
What  general  principles  are  used  in  finding  the  greatest  common 
divisor?  (54.)  Give  the  rule  for  finding  the  greatest  common 
divisor  by  the  method  of  factors.  (55.)  What  additional  prin- 
ciple is  used  ?  (56.)  Give  the  rule  for  finding  the  greatest  com- 
mon divisor  by  the  method  of  continued  division.  How  do  you 
find  the  greatest  common  divisor  of  more  than  two  numbers? 
(57.)  What  is  a  multiple  of  a  number?  What  is  a  common 
multiple  of  two  or  more  numbers?  What  is  their  least  common 
multiple?  (58.)  Give  the  principles  used  in  finding  the  least 
common  multiple.  (59.)  Give  the  rule  for  finding  the  least 
common  multiple. 


COMMON    FRACTIONS. 

DEFINITIONS. 

60.  If  a  unit  is  divided  into  equal  parts, 
each  part  is  called  a  fractional  unit. 
If  the  unit  is  divided  into  two  equal  parts, 
each  is  called  one-half ;  if  into  three,  each  is  called  one- 
third  ;  if  into  four,  each  is  called  one-fourth  ;  and  so  on. 
Fractional  units   may  be  written  and  read  as  shown 
below: 

h  h  h  i.  h  h         etc. 

one-half,     one-third,    one-fourth,    one-fifth,    one-sixth,  one-seventh,     etc. 

The  Reciprocal  of  a  Number  is  1  divided  by  that 
number.  Thus,  ^  is  the  reciprocal  of  2,  -J-  is  the  reciprocal 
of  3,  and  so  on. 

MENTAL     EXERCISES. 

1.  If  a  unit  is  divided  into  5  equal  parts,  what  is  each 
part  called  ?  If  it  is  divided  into  9  equal  parts,  what  is 
each  part  called  ?     If  into  10  ?    If  into  12  ?    If  into  13  ? 

2.  How  many  halves  of  an  apple  are  there  in  1  apple  ? 
How  many  fifths?  How  many  ninths?  How  many 
tenths?    KowmgLny  twelfths?    Fifteenths?    Twentieths? 


REDUCTION.  81 

3.  What  is  the  reciprocal  of  10  ?  Of  what  number  is 
tV  the  reciprocal  ?    ^^V  ?    j\?     ^h?     gV  •' 

61.  A  Fraction  is  a  fractional  unit,  or  a  collection  of 
fractional  units ;  thus,  one-half,  two-thirds,  four-7iinths, 
etc.,  are  fractions. 

Fractions  may  be  written  and  read  as  shown  below  : 

h  h  h  A,        etc. 

three-fourths,       two-ninths,       five-sevenths,         eight-elevenths,    etc. 

Fractions  written  in  this  manner  are  called  vulgar,  or 
Common  Fractions. 

Common  fractions  are  expressed  by  two  numbers,  one 
written  above  the  other,  with  a  line  between  them.  The 
number  below  the  line  is  called  the  Denominator,  and 
the  one  above  it  is  called  the  Numerator.  Both  numera- 
tor and  denominator  are  called  Terms  of  the  fraction. 

The  denominator  indicates  the  number  of  equal  parts 
into  which  the  unit  is  divided,  and  the  numerator  shows 
how  many  of  these  parts  are  taken.  Thus,  in  the  fraction 
I,  the  denominator  indicates  that  1  is  divided  into  4  equal 
parts,  and  the  numerator  shows  that  3  of  these  are  taken. 

MENTAL      EXERCISES. 

1.  If  1  is  divided  into  7  equal  parts,  what  is  1  part 
called  ?  What  are  4  parts  called  ?  If  1  is  divided  into 
11  equal  parts,  what  are  5  of  them  called  ?  6  of  them  ? 

2.  If  1  yard  is  divided  into  8  equal  parts,  what  is  1  part 
called  ?  3  parts  ?  5  parts  ?  7  parts  ?  How  many 
eighths  of  an  apple  are  there  in  1  apple  9  in  3  apples  ?  in 
7  apples  9  How  many  elevenths  in  6  ?  in  11  ?  in  13  ? 
in  17  ?  in  19  ?  How  many  twelfths  in  5  ?  in  11  ?  in  13  ? 
in  15  ?   in  21  ? 


83  COMMON     FRACTIONS. 

3.  In  the  fraction  |,  what  is  the  denominator  ?  What 
is  the  numerator  ?  What  is  the  fractional  unit  9  How 
many  times  is  this  unit  taken  ?    How  many  times  ^  are  -|  ? 

Write  the  following  fractions : 

1.  Seyen-eighths.  4.  Eleven-hundredths. 

2.  Four-tenths.  5.  Thirteen-twenty  fifths. 

3.  Mne-twentieths.         6.  Sixty-thousandths. 
Kead  the  following  fractions : 

!•    T^    11?  TT'  Yty  "36?    10  0?    1000?  ¥5?  TTT?  tI?  IT?  TT* 
'^^    f ?  "2^?  "STO?    1000?  TTF?  "BTT?  IT?  "STt?  TOOTT?  Wo?   29T' 

63.  A  fraction  is  equal  to  its  numerator  divided  by- 
its  denominator.  Thus,  f  is  equal  to  3-^4;  for,  if 
each  of  the  units  in  3  is  divided  into  four  equal  parts,  we 
shall  have  12  such  parts,  each  equal  to  J,  that  is,  we  shall 
have  12  fourths  ;  but  12  fourths  divided  by  4  is  equal  to 
3  fourths,  or  to  f ,  that  is,  3  -f-  4  is  equal  to  J. 

63.  A  fraction  may  be  regarded  either  as  a  number,  or 
as  an  indicated  division  : 

1°.  Eegarded  as  a  number,  the  unit  is  fractional  and 
equal  to  the  reciprocal  of  the  denominator.  Thus,  f  is  a 
collection  of  3  units,  each  equal  to  -J-,  that  is,  f  =  3  x  i. 

2°.  Regarded  as  an  indicated  division,  the  numerator  is 
the  dividend  and  the  denominator  is  the  divisor.  Thus, 
1  =  3-^4. 

MENTAL     EXERCISES. 

1.  In  the  fraction  %-^,  what  is  the  fractional  unit  9 
How  many  times  is  it  taken  ?    What  is  7  times  %^  ? 

2.  Is  there  any  difference  between  A  of  1  pound  and  -^ 
of  9  pounds  ?  4  <^f  1  dollar,  is  what  part  of  4  dollars  ? 
How  does  f  of  1  dollar  differ  from  }  of  4  dollars  9 


REDUCTION.  83 

DEFINITIONS. 

64.  A  Proper  Fraction  is  one  in  which  the  numera- 
tor is  less  than  the  denominator ;  as,  J,  f . 

An  Improper  Fraction  is  one  in  which  the  numera- 
tor is  equal  to,  or  greater  than  the  denominator ;  as,  f ,  f . 

Note. — If  the  numerator  is  less  than  the  denominator,  the  frac- 
tion is  less  than  1 ;  if  the  numerator  is  equal  to  the  denominator,  the 
fraction  is  equal  to  1 ;  if  the  numerator  is  greater  than  the  denom- 
inator, the  fraction  is  greater  than  1. 

A  Simple  Fraction  is  one,  both  of  whose  terms  are 
whole  numbers ;  as,  f ,  f . 

A  Mixed  Number  is  a  number  composed  of  an  in- 
tegral and  of  a  fractional  part ;  as,  2^,  S-iJ-. 

A  Complex  Fraction  is  one  that  has  at  least  one  of 

its  terms  fractional ;  as,  ^,  zrjr,   tjt,   qI. 

Note. — A  whole  number  may  be  regarded  as  a  fraction  whose 
denominator  is  1.    Thus,  8  =  f . 

FUNDAMENTAL     PRINCIPLES. 

65.  Because  the  numerator  shows  how  many  times  the 
fractional  unit  is  taken,  we  have  the  following  principles: 

1°.  Multiplying  the  numerator  of  a  fraction  hy  any 
number  is  equivalent  to  7nultiplyi7ig  the  fraction  ly  that 
nurriber, 

2°.  Dividing  the  numerator  of  a  fraction  hy  any  nu7n- 
her  is  equivalent  to  dividing  the  fraction  by  that  number. 

Because  the  denominator  shows  the   number  of  equal 
parts  into  which  we  divide  the  unit  1  to  obtain  the  frac- 
tional unit,  we  have  the  following  principles  : 
.    3°.  Multiplying  the  denominator  of  a  fraction  by  any 


84  COMMON     FEACTIONS. 

number  is  equivalent  to  dividitig  the  fraction  by  that  num- 
ber. 

4°.  Dividing  the  denominator  of  a  fraction  by  any  num- 
ber is  equivalent  to  muUiplyi7ig  the  fraction  by  that  num- 
ber. 

From  principles  1°  and  3°,  and  from  principles  2°  and 
4°,  we  have  the  following  principles : 

5°.  Multiplying  both  terms  of  a  fraction  by  the  same 
number  does  not  alter  its  value. 

6°.  Dividing  both  terms  of  a  fraction  by  the  same  num- 
ber does  not  alter  its  value. 

REDUCTION     OF     FRACTIONS. 

66.  Reduction  is  the  operation  of  changing  the  form 

of  a  number  without  altering  its  value. 

The  methods  of  reducing  fractions  depend  on  the  principles  just 
deduced. 

67.  To  reduce  a  whole  number  to  the  form  of  a 
fraction  having  a  given  fractional  unit. 

MENTAL      EXERCISES. 

1.  In  4  apjjles,  how  many  tenths  of  an  apple  ? 

Explanation. — In  1  apple  there  are  10  tenths,  hence  in  4  apples 
there  are  4  times  10,  or  40  tenths. 

2.  How  many  quarters  of  a  dollar  must  I  pay  for  a  hat 
worth  $4  ?    How  many  quarters  in  4  ?  in  5  ? 

3.  How  many  sevenths  are  there  in  12?  in  15?  in  17? 
in  30  ?  What  is  the  difference  between  12  and  -^  ?  be- 
tween 15  and  ^^  ?    What  is  the  fractional  unit  of  J-^  ? 

4.  How  many  tenths  in  9  ?  in  17  ?  How  many  elevenths 
in  11  ?     How  many  in  13  ?  in  17  ?  in  19  ? 


REDUCTION.  85 

Let  it  be  required  to  change  17  to  the  form  of  a  frac- 
tion whose  unit  is  -J- : 

Explanation. — Having  writ- 

ir,  J         ^1.       *  *  -P  OPERATION. 

ten  17  under  the  form  of  a  frac- 

tion,  (Art.  64),  we  multiply  both  ^7  _-  £[  _  iL^  —  i!^. 

of  its  terms  by  6,  (Prin.  5°),  which  1  1x6  6  * 

gives  ^K 

In  like  manner  we  may  treat  all  similar  cases  ;  hence,  the  fol- 
lowing 

RULE. 

Multiply  the  number  by  the  denominator  of  the 
given  unit  and  write  the  product  over  that  denomi- 
nator. 

EXAM  PLES. 

Reduce  Reduce 

1.  12  to  the  unit  \.  6.     59  to  the  unit  ^. 

2.  14  to  the  unit  \.  7.  212  to  the  unit  ^. 

3.  7  to  the  unit  \.  8.  524  to  the  unit  ■^. 

4.  19  to  the  unit  \.  9.  326  to  the  unit  ^. 

5.  42  to  the  unit  ^.  10.  426  to  the  unit  -^. 
68.  To  reduce  a  mixed  number  to  the  form  of  a 

simple  fraction. 

MENTAL     EXERCISES. 

1.  In  ^\  pounds,  how  many  fourths  of  a  pound  9 

Explanation. — In  S  pounds  there  are  3  x  4,  or  12  fourths  of  a 
pound,  hence  in  4f  pounds  there  are  12  +  3  or  15  fourths  of  a  pound. 

2.  How  many  tenths  of  a  dollar  must  I  give  for  a  book 
worth  $3^?  How  many  tenths  in  5^3^?  in  9^?  in 
11^  ?    How  many  tioelfths  in  7^  ?  in  13 A  ? 

3.  How  many  ninths  are  there  in  5f  ?  What  is  the 
difference  between  5|  and  ^^?  What  is  the  fractional 
unit  of  Y?     How  many  fifteenths  in  7  ?  in  13  ? 


86  COMMON     FRACTIONS. 

Let  it  be  required  to  reduce  12f  to  the  form  of  a  simple 
fraction : 

Explanation. — The  mixed  num-  operation. 

ber  12 1  is  equal  to  12  +  ?  ;  reducing  3        84       3        87 

its  integral  part  to  tlie  unit  i,  it  be-         l'^  +  iv  ==^  ~w"  +  ^^  =  —  • 
comes   V,  th^t  is,  the  given  num- 
ber is  equal  to  ^ ,  taken  84  +  3  times,  or  to  -\\ 

In  like  manner  we  may  treat  all  similar  cases  ;  hence,  the  fol- 
lowing 

RULE. 

Multiply  the  entire  part  by  the  denojninator  of 

the  fraction  and  to  the  product  add  the  numerator ; 

then  place  the  sum  over  the  given  denominator. 

EXAM  PLES. 

Keduce  the  following  numbers  to  simple  fractions: 


1.3^. 

15.  27*. 

29.  152ff. 

2.  ^. 

16.  15*. 

30.  207if. 

3.  lOf 

17.  44*. 

31.  3914f 

4.  12^. 

•    18.  31*. 

32.  2373VJ. 

5.  15^. 

19.  22*. 

33.  215^1. 

6.  6f|. 

20.  100*. 

34.  187i|. 

7.  lOlf 

21.  102* 

35.  1,6303^. 

8.  64|. 

22.  25*. 

36.  579*»^. 

9.  15*. 

23.  31f 

37.  4,311if. 

10.  123%. 

24.  118-H. 

38.  3,204|f. 

11.  $19*. 

25.  %ll^. 

39.  n^lbs. 

12.  164-/^*5. 

26.  I316|. 

40.  472*  yds. 

13.  A.^yds. 

27.  $177*.  _ 

41.  3651  r/«. 

14.  S^1lrs, 

28.  210J  lbs. 

42.  $290*. 

69.  To  reduce 

an  improper  fraction  to  the  form 

of  an  integral  or  a  mixed  number. 

REDUCTIOKo  87 

MENTAL     EXERCISES. 

1.  In  11  ffths  of  a  pound,  how  msmj  pounds  and  parts 

of  a  pound  ? 

Explanation. —  Because  5  fiftJis  of  a  pound  make  1  pound, 
17  fifths  will  make  ^^-,  or  d^  pouTids. 

2.  In  4^  of  a  dollar,  how  many  dollars  and  how  many 
tenths  of  a  dollar  9  In  ^  of  a  yard,  how  many  yards  and 
how  m'dnj  fourths  of  a  yai^d  ?  How  many  whole  feet  and 
what  fraction  of  a  foot  are  there  in  ^^  of  a  foot  ? 

3.  A  boy  bought  18  melons  at  $J-  a  piece ;  what  did  he 
pay  for  them  ?  What  would  they  have  cost  at  $^  each  ? 
at  $i?  at  %?  at  $|?  at  $|?  at  l-^-?  at  $^  ? 

Let  it  be  required  to  reduce  -^  to  a  mixed  number. 

Explanation. — We  perform  the  indicated  opbbation. 

division,  that  is,  we  divide  the  numerator  by  22)10'4(4i4 

the  denominator  and  find  the  quotient,  4|f,  ^^ 

which  is  a  mixed  number.  Z— 

In  like  manner  we  may  treat  all  similar  16   Bern. 
cases ;  hence,  the  following 

RULE. 

Divide  the  numerator  by  the  denoTninator ;   the 

quotient  will  he  the  required  number. 

Note. — If  the  remainder  is  0,  the  division  is  exact  and  the  given 
fraction  is  a  whole  number  under  a  fractional  form. 

EXAMPLES. 

Reduce  the  following  fractions  to  mixed  numbers: 

1.  V.  5.  W-  9-  ^m- 

2.  x^.  6.  W-  10.  -Vtf. 


88  COMMON     FKACTIONS. 

13.  frt.  19.  ^^^.        25.  miK 

14.  ^/.  30.   ^^.  26.  ^MF. 

15.  ^.  21.  HF.  27.  ^U^. 

16.  i#.  22.  ^^.  28.  -\%W-' 

17.  J^t  23.  -4^1^.  29.  -WA'-. 

18.  W^.  24.  ^ItfA.  30.  ^»|ii. 

70.   To  reduce  a  fraction  to  its  lowest  terms. 

A  fraction  is  said  to  be  in  its  lowest  terms  when  its 
terms  are  prime  with  respect  to  each  other,  that  is,  when 
they  have  no  common  factor. 

MENTAL      EXERCISES. 

1.  If  \  is  divided  into  3  equal  parts,  what  is  the  value 
of  each  part  ?  How  many  thirds  are  there  in  3  ninths  ? 
in  G  tiinths  f  How  does  f  of  an  apple  differ  from  -J  of 
an  apple?  Is  there  any  difference  between  the  values 
i  and  I  ?  of  ^5^  and  f  ?  of  ^^  and  |? 

2.  If  we  divide  the  dividend  and  divisor  by  the  same 
number,  what  effect  will  it  have  on  the  quotient  ?  If  we 
divide  both  terms  of  a  fraction  by  the  same  number,  will 
it  affect  the  value  of  the  fraction  ? 

Let  it  be  required  to  reduce  the  fraction  ^^  to  its  low- 
est terms. 

Explanation. — We  first  resolve  the  opebatiok. 

terms  of  the  fraction  into  prime  fac-  30         2  .  $  .  $       2 

tors,  and  then   strike   out  those  that  Jq^  ^^  *     rf    w  =  1^ 

are   common    to    both,   (Principle  6°, 

Art.  05) ;  the  resulting  fraction  is  equivalent  to  the  given  one,  and 
is  in  its  lowest  terms. 

Since  all  similar  cases  may  be  treated  in  like  manner,  we  have 
the  following 


EEDUCTION.  89 

RULE. 
Resolve  the  terms  of  the  fraction  into  prime  fac- 
tors, and  cancel  all  that  are  common  to  both. 

Note, — If  the  terms  cannot  be  factored  by  inspection,  find  theii 
greatest  common  divisor  and  divide  them  both  by  it. 

EXAM  PLES. 

Reduce  the  following  fractions  to  their  lowest  terms : 

1.  If.  14.  AV  27.   Htt- 

AM, 


2.  T»^.  15.  ^.  38.  ^„ 

3.  Ml-  16-  tUt-  39.  m^ 

4.  mi-  17-   m-  30.  ftif 

5.  m.  18.  m-  31.  \m 

6.  1^.  19.  T%:.  33.  -JJSA 


7.  m-  20.  m-  33.  mh 

8-  AV-  21-  W-  34.  mi.. 

9.  «?.  33.  im-  35.  fH4- 

10.  Jtt.  33.  m%-  36.  T^ftftlt- 

11.  m-  34.  IK-  87.  «M- 

12.  m-  35.  IMl-  38.  ««. 

13.  iM.  •  36.  ,%iV-  39.  ilff. 

71.  To  reduce  a  fraction  to  an  equivalent  frac- 
tion whose  denominator  is  a  multiple  of  the  given 
denominator. 

Let  it  be  required  to  reduce  f  to  an  equivalent  fraction 

wlieu  the  denominator  is  12 : 

Explanation. —  Here  we  have  mul-  operation. 

tiplied  both  terms  of  the  given  fraction  3       3x3         9 

by  V"'  ^^  ^  '  *b^  resulting  fraction,  j\,  is  ^  ^  4~X~3        \% 

equivalent  to  tlie  given  fraction  (Prin.  5°), 
and  its  denominator  is  12,  tliat  is,  its  fractional  unit  is  yV. 


90  COMMON     FRACTIONS. 

In  like  manner  other  fractions  may  be  reduced  ;  hence,  the 

RULE. 

Multiply  hoth  terms  of  the  fraction  hy  the  quo- 
tient of  the  required  denominator  divided  hy  the 
given  denominator. 

EXAMPLES. 

1.  Eeduce  f  to  a  fraction  whose  unit  is  -g^. 

Arts  ?_^-l? 
Eeduce,  *  5  x  6  ~"  30* 

2.  -f^  to  the  unit  ■^.  6.  ^  to  the  unit  -g^. 

3.  j^  to  the  unit  -^.  7.  ^  to  the  unit  ^1^. 

4.  -^  to  the  unit  yfg-.  8.  -^  to  the  unit  ^\^. 

5.  ^  to  the  unit  y^q-  ^*  A  ^^  ^^^  unit  y^. 

72.  To  reduce  two  or  more  fractions  to  equiva- 
lent fractions  having  a  common  denominator. 

Let  it  be  required  to  reduce  f,  f,  and  f  to  a  common, 
denominator ; 

Explanation, — Here  we  have 
multiplied  both  terms  of  each  frac- 
tion by  the  product  of  the  denom- 
inators of  the  other  fractions ;  the 
denominators  of  each  of  the  result- 
ing fractions  is  then  equal  to  the 
continued  product  of  the  denomina- 
tors of  all  the  fractions. 

In  like  manner  we  may  treat  all  other  groups  of  fractions 
hence,  the  following 

RULE. 
Multiply  hoth  terms  of  each  fraction  hy  the  pro- 
duct of  the  denominators  of  all  the  other  fractions. 


OPERATION. 

2 

2  X  ,4  X  7      56 

3' 

~  3  X  4  X  7  ~  84' 

3 

3x3x7      63 

4' 

4  X  3  X  7  ~  84' 

2 
7' 

2x3x4      24 
~  7  X  3  X  4  ~  84' 

REDUCTIOir.  91 

Note. — If  the  denominators  are  small,  the  operation  can  be  per- 
formed mentally, 

EXAM  PLES. 

Reduce  to  a  common  denominator : 

1.  i,  f ,  and  |.  6.  -^,  ^,  and  fl- 

2.  I  },  and  t.  7.  {^,  ||,  and  A. 

3.  f,  f,  and  f .  8.  ^,  H,  and  H- 

4.  A.  A,  and  t3^.  9.  f ,  |,  and  IJ. 

5.  M>  H,  and  A.  10.  f,  A.  and  |f . 

73.  To  reduce  two  or  more  fractions  to  their 
least  common  denominator. 

Let  it  be  required  to  reduce  -^,  |f ,  and  |-J  to  equivalent 
fractions  having  the  least  possible  common  denominator. 

Explanation.— Here  we  first  opebatiok. 

reduce  each  fraction  to  its  low-  8  2  2x8       16 

est  terms  (Art.  70) ;   we  next         12  ^^  S   ^^    o  ^  o  ^^  94' 
find  the  least  common  multiple, 

24,  of    the  new  denominators,         15         3  3x6        18 

which  will  be  the  required  de-         20         4  4  ^  ^  —  24' 

nominator ;  we  then  divide  this 

multiple  by  each  denominator         20 5   5x2 10 

separately,   and  multiply  both         48        12        12  X  2        24' 
terms  of  the  corresponding  frac- 
tion by  the  quotient.     Thus,  we  multiply  both  terms  of  |  by  8, 
both  terms  of  J  by  6,  and  both  terms  of  -^-^  by  2. 

In  like  manner  we  may  treat  all  similar  cases  ;  hence,  the  fol- 
lowing 

RULE. 

I.  Reduce  each  fraction  to  its  simplest  terms. 

II.  Find  the  least  common  multiple  of  all  the 
denominators  for  a  common  denominator;  divide 
this  by  each  denominator  separately  and  multiply 
the  corresponding  numerator  by  the  quotient. 


92  COMMON^     FRACTIONS. 

EXAM  PLES. 

Eeduce  each  of  the  following  groups  of  fractions  to  a 
common  denominator : 

1.  h  h  i>  and  T^.  Ans.  J|,  Jf,  fi  and  4|. 

2.  I,  I,  },  and  ^^.  Ans,  yVo.  ^.  t¥o.  and  y^-. 

3.  f,  T^,  If,  and  A.  17.  W.  M.  and  «• 

4.  i  A,  H,  and  •^.  18.  HJ,  ^V,  and  ^^3. 
^-  A»  A»  yt^  and  -^.           19.  f,  f,  y^-,  and  f|. 

6.  f,  tt.  and  If.  20.  «,  A.  ft.  and  JJf . 

7.  if,  if,  and  tt-  21.  i,  i  h  h  and  f . 

8.  J^,  1,  and  A.  22.  f ,  ^s^,  A,  A,  and  if 

9.  t3^,  t^,  and  ff .  23.  A.  M.  W'  and  ^. 

10.  tt,  A,  and  a-  24.  W,  t¥o,  U,  and  if. 

11.  i^,  A,  and  i«.  25.  if,  ^,  i^i^,  and  li. 

12.  A.  ii  and  «.  26.  ^,  ^,  ii,  and  iff. 

13.  A.  tIt,  and  ^f^.  27.  f^,  f f ,  ^VV,  and  -W- 

14.  A,  ^,  and  A.  28.  4|,  f  J,  if,  and  -,f^. 

15.  H,  T^g,  and  ■^.  29.  5f,  11^,  ^,  and  J^- 

16.  fi,  if,  and  e.  30.  6f,  llf,  4^,  and  ff. 

Note  1.— If  there  are  any  Integral,  or  any  mixed  numbers,  reduce  them  to  the 
form  of  simple  fractions  (Arts.  67  and  68). 

Note  2.— Complex  fractions  are  reduced  to  simple  ones  by  the  rule  for  division 
of  fractions  (Art.  81). 

ADDITION    OF    FRACTIONS. 
DEFINITION. 

14.  Addition  of  Fractions  is  the  operation  of  find- 
ing the  sum  of  two  or  more  fractions. 

MENTAL     EXERCISES. 

1.  What  is  the  fractional  unit  of  ^  of  $1  ?  of  ^6^.  of  $1  ? 
How  many  times  is  this  unit  taken  in  $^  and  $r^  ? 


ADDITION.  93 

2.  What  is  the  sum  of  -^  and  -^  ?  of  ■^,  y\,  and  -^  ? 
of  A.  A.  A  and  A  ?  of  A,  A,  H,  and  ^  ? 

3.  How  many  twelfths  are  there  in  f?  How  many 
twelfths  in  f  ?  What  is  the  sum  -^  and  ^  ?  What  is  the 
sum  of  f  and  |  ?     What  is  the  sum  of  f  and  f  ? 

Note. — The  fractions  must  be  reduced  to  the  same  fractional 
unit  before  they  can  be  added,  that  is,  they  must  be  reduced  to  a 
common  denominator, 

4.  A  man  bought  ^  of  a  pound  of  indigo  at  one  time 
and  I  of  a  pound  at  another  time  ?  how  mucli  did  he  buy 
in  all  ?    What  is  the  sum  of  J  and  f  ?  of  f  and  |  ? 

5.  What  is  the  sum  of  $4^  and  $2^  ?  of  ^  and  J? 
of  ^  and  ff  ?    What  mixed  number  is  equal  to  ||  ? 

6.  What  is  the  sum  of  $|,  1^,  and  1^?  of  |,  \,  and  3^? 
of  IJ,  JV?  and  ^?    What  mixed  number  is  equal  to  If^? 

OPERATION    OF    ADDITION. 

75.   Let  it  be  required  to  find  the  sum  of  f  and  -f . 

Explanation.  —  Having  operation. 

reduced  the   fractions  to  a  4      3       28      15       43 

common    denominator,    we  5      7       35      35  ^^  35  ^^         * 

see  that  the  first  is  equal  to 

the  fractional  unit  3^5  taken  28  times,  and  the  second  to  the  same 
unit  taken  15  times  ;  hence,  the  sum  is  equal  to  this  unit  taken 
38  +  15,  or  43  times,  that  is,  to  f  f  or  to  Ig^. 

In  like  manner  we  may  treat  all  similar  cases ;  hence,  the  fol- 
lowing 

RULE. 

/.  Reduce  the  fractions  to  simple  fractions  having 
a  common  denominator. 

II.  Add  their  numerators  for  a  new  numerator, 
and  write  the  sum  over  the  common  denominator. 


94  COMMON     FRACTIOiJS. 

MENTAL     EXERCISES. 

1.  What  is  the  sum  of  |y\,  $^,  and  IJ|  ?    Ans.  |1|. 
Add  the  following  groups  of  fractions  : 

2.  I,  I  and  t'j.  9.  |,  Jf  ,  and  J#. 

3.  I,  f ,  and  i.  10.  I,  tV  and  -W. 

4.  I,  f,  and  T^V.  11.  i,  ^,  h  and  «. 

5.  i  A.  and  A-  12-  f,  A,  H.  and  f . 

6.  A,  i  ^,  and  J.  13.  |,  ^,  i,  |,  and  |. 

7.  f,  4,  and  ^.  14.  ^\,  J,  |,  4^,  and  J. 

8.  J,  J,  i,  and  i.  15.  +,  -ft-,  f,  -g^,  and  i- 

.     When  there  are  mixed  numbers,  add  the  sum  of  the  fractional 
parts  to  the  sum  of  the  whole  numbers. 

16.  4^,  6J,  2i,  and  |.  Ans.  12  +  ff  =  13^- 

17.  10|,  7i,  8|,  and  16}.  Ans.  42^. 

18.  1-^,  6f,  18if ,  and  2^.  Ans.  28^^. 

19.  2^,  6^,  and  12H-  30.  2^,  5^,  and  6^. 

20.  67^^,  4^,  and  600f.  31.  6^,  4|,  and  13|. 

21.  13^-,  99|,  and  512-jV  32.  6f,  llt^j,  and  9. 

22.  14t^,  d^,  and  88^-  33.  {i,  -^,  and  2^- 

23.  900^,  450^%,  and  6^.  34.  ^,  ll-fj,  and  5f. 

24.  21i-,  98^^,  and  14^^.  35.  18-J-,  2^^,  and  6J. 

25.  li,  4^,  and  6|.  36.  ^,  ^,  and  8^. 

26.  ^,  7tV,  and  8^.  37.  9^,  ||,  and  «. 

27.  ^,  ^S,  and  7^^.  38.  5f  3-^,  and  «. 

28.  5i,  i  and  7^.  39.  16^  |i,  and  fj. 

29.  4|,  3^,  and  3^.  40.  8^,  «,  and  f|. 
■41.  $1  +  $4  +  1^  +  $1  +  $3i  z=  ? 

42.  37i?J5.  +  24|/^'5.  +  ^/5.  +  4ii-ZJ5.  =  ? 

43.  46}  +  118f  +  319|  +  1^  =  ? 

44.  ^^yds.  +  m^yds.  +  l^\yds.  +  82tt«/«^6^  =  ? 


ADDITION.  95 

PRACTICAL     PROBLEMS. 

1.  A  farmer  has  3  fields ;  the  first  contains  31-|  acres, 
the  second  49f  acres,  and  the  third  59^  acres :  how  many 
acres  has  he  in  all  ?  Ans.  14:0^^  Acres, 

2.  A  man  earned  $3|  the  first  day,  $4|  the  second  day, 
15^  the  third  day,  $7i  the  fourth  day,  $4^  the  fifth  day, 
and  $3f  the  sixth  day ;  how  much  did  he  earn  in  the  six 
days  ?  Ans.  $28f. 

3.  A.  traveled  17f  miles  the  first  day,  f  of  17f  miles  the 
second  day,  22-J  iniles  the  third  day,  and  36^-  77iiles  the 
fourth  day ;  how  far  did  he  travel  in  the  four  days  ? 

4.  B.  works  S^Jiours  on  Monday,  9^hot(rs  on  Tuesday, 
S^Q  liours  on  Wednesday,  lOj  hours  on  Thursday,  9^  hours 
on  Friday,  and  10^  hours  on  Saturday ;  how  many  hours 
does  he  work  during  the  week  ? 

5.  A  farmer  sells  3  tons  of  hay  for  147.  ISf,  3  cows  for 
$111,421,  a  horse  for  ^173.16|,  and  100  hushels  of  oats  for 
$62.87|^;  how  much  does  he  receive  for  all  ? 

6.  How  many  dollars  will  pay  for  a  coat  worth  $14},  a 
hat  worth  %b^,  a  vest  worth  %Q\,  a  pair  of  pants  worth  $8, 
and  a  pair  of  boots  worth  $9^  ? 

7.  How  many  pounds  of  butter  in  4  tubs  weighing 
respectively  27^/^',^.,  34|/^*\,  ^^lhs.,  and  29f  Z^>s.? 

8.  How  many  tons  of  coal  in  5  loads  weighing  respec- 
tively li,  lA,  lA.  H.  and  11  tons  9 

9.  How  many  yards  in  4  pieces  of  cloth,  measuring 
respectively  27^yd-<^.,  37f //r/,^.,  39^ yds.,  and  30{^  yds.? 

10.  A  farm  contains  26^  acres  of  plough  land,  39^  acres 
of  wood  land,  61f  acres  of  pasture  land,  and  42^  acres  of 
meadow  land ;  how  many  acres  in  the  farm  ? 


96  COMMON     FEACTIONS. 

SUBTRACTION     OF     FRACTIONS. 
DEFINITIONS. 

76.  Subtraction  of  Fractions  is  the  operation  of 
finding  the  difference  between  two  fractions. 

MENTAL      EXERCISES. 

1.  "What  is  the  difference  between  f  and  f  ?  ^  and  ^  ? 
^and^%,?    MandH?    Mand|4?    ii  and  ^  ? 

2.  .What  is  the  difference  between  |-  and  -f? 

Explanation. — The  minuend  is  equal  to  f^  and  the  subtrahend 
is  equal  to  i| ;  the  difference  is  therefore  equal  to  the  fractional  unit 
-^Q  taken,  25  —  12,  or  13  times,  that  is,  it  is  to  ^f. 

3.  A  boy  had  -J  of  a  dollar,  but  he  spent  -^  of  a  dollar 
for  a  slate  ;  how  much  had  he  left  ?  What  is  the  differ- 
ence between  -J  and  ^  ?    ||-  and  ^  ? 

4.  -f-  -  I  =  ?  6.  ^Ibs.  -ilb.  =  ? 

5.  $1^  -$^=?        7.  2^ft.  -  {iff.  =  ? 

OPERATION      OF     SUBTRACTION. 

77.  Let  it  be  required  to  find  the  difference  between 
I  and  f . 

Explanation. — Having  reduced 
the  given  fractions  to  the  common 
unit,  -^^,  we  see  that  the  minuend 
contains  this  unit  35  times  and  the 
subtrahend  16  times  ;  hence,  the  re- 
mainder contains  it  35  —  16,  or  19  times ;  the  remainder  is  there- 
fore H. 

Since  all  similar  examples  may  be  treated  in  the  same  manner,  we 
have  the  following 

RULE. 

/.  Reduce  the  fractions  to  simple  fractions  having 

a  common  denominator. 


r. 

5 

8" 

7 

35 
"56 

16 
56^ 

19 
"56* 

SUBTRACTION.  9*? 

//.  Subtract  the  nuinerator  of  the  subtrahend 
from  that  of  the  minuend  for  a  ne^w  numerator, 
and  write  the  difference  over  the  common  denomi- 
nator. 

EXAMPLES. 

Find  the  difference  between, 


1.  «andtt. 

17.  $18i|  and  $13^. 

2.  A  and  A. 

18.  14^ ft.  and  l^ft. 

3.  37^  and  33^. 

19.  lOm  and  4^^. 

4.  ^  and  4J. 

20.  206^^6/5.  and  194f  ^^^5. 

6.  13^^^  and  9^. 

21.  4ni\yds.  and  21^  yds. 

6.  SO-jij  and  47^. 

22.  $11811-  and  |24|4. 

7.  42  and  30t^. 

23.  $22-3!^  and  $9-3%. 

8.  90^  and  25^. 

24.  2463^5-  and  194f. 

9.  46-1  and  15J. 

25.  l,476ff  and  894^. 

10.  T^  and  f  off. 

26.  177f^'^^.and66H^i*. 

11.  98|-  and  45f. 

27.  163|??2f.  andl08|wi. 

12.  150^  and  65^1,^. 

28.  $864iV  and  $648|. 

13.  ^  and  f 

29.  146|^owsand97^^ow5. 

14.  ^^  and  If. 

30.  %%1^  acres  Q^ndiUll  acres. 

15.  134  and  7^. 

31.  1,884t^w.  and  l,801^m. 

16.  843ff  and  94^. 

32.  ^^O^rods  and  199^. ro^5. 

33.  $180^  H-  $27^  - 

-  ($431  +  I39|)  =  ? 

34.  146J  rods  +  73 J  rods  —  (24^  rods  —  6^  rods)  =  ? 

35.  nShu.  +  375|^.«^.  -  (46  5w.  ~  7i  Jw.)  =  ? 

PRACTICAL     PROBLEMS. 

1.  A.  has  $4^,  and  B.  has  $d^ ;  how  much  more  has 
A.  than  B.  ?  Ans.  I  J. 

2.  A.  bought  bQ^ pounds  of  butter,  and  sold  ^  of  it  to 

4 


98  COMMON     FRACTIONS. 

one  person  and  13^ pounds  to  another ;  how  much  had  he 
left?  Ans,  14i\l^  pounds, 

3.  A  grocer  bought  2  hogsheads  of  sugar,  each  weigh- 
ing 1,302  2J02inds ;  he  sold  -J  of  one  hogshead  and  455^ 
pounds  from  the  other :  how  many  pounds  had  he  remain- 
ing ?  Ans.  1,714^  ?5s. 

4.  A  merchant  bought  2  pieces  of  cloth  ;  the  first  con- 
tained 3^ yards,  and  the  second  414^ yards;  he  then  sold 
69^  yards :  how  many  yards  had  he  left  ? 

5.  A  man  bought  a  farm  of  211|  acres,  and  sold  llY^Sj. 
acres  of  it ;  how  much  had  he  remaining  ? 

6.  From  a  cask  containing  60 1  gallons  of  cider  there  were 
drawn  off  Yt\^ gallons  ;  how  much  was  there  left  ? 

7.  A  sloop  has  on  board  406|  tons  of  coal,  of  which  311f 
tons  is  anthracite,  and  the  remainder  cannel ;  how  much 
cannel  does  she  contain  ? 

8.  A  merchant  had  a  piece  of  silk  containing  4:2^  yards, 
from  which  he  sold  17|-  yards  j  how  much  had  he  remain- 
ing? 

9.  A  person  having  $49f,  spent  I4f  in  getting  to  Bos- 
ton, and  $5^  in  getting  to  Portland ;  how  much  had  he 
left  on  reaching  Portland  ? 

10.  If  a  man  has  $11^  and  spends  I8f,  how  much  will 
he  have  left  ? 

11.  A  tailor  had  a  piece  of  cloth  containing  2^ydsj  he 
cut  off  4J-  yds.  to  make  a  coat  and  If  yds.  to  make  a  pair 
of  pants :  how  many  yards  were  left  ? 

12.  A  man  had  to  walk  dill  miles;  ^®  walked  30^  miles 
the  first  day,  33|  miles  the  second  day,  and  finished  the 
journey  the  third  day :  how  far  did  he  walk  the  third  day  ? 


MULTIPLICATION.  99 

13.  A  cask  of  wine  contained  42 1^  gallo7is ;  of  this 
13^ gallons  were  drawn  off,  and  12^ gallons  leaked  out: 
how  much  remained  in  the  cask  ? 

14.  A  grocer  bought  Sd^lbs.  of  tea;  of  this  he  sokl 
13|-  lbs.  to  one  customer,  9^  lbs.  to  a  second  customer,  and 
VZ^lbs.  to  a  third  customer :  how  many  pounds  had  he  left  ? 

15.  A  laborer  earned  $18|  and  received  as  a  gift  $14^; 
he  then  bought  a  barrel  of  flour  for  $12,  and  groceries  to 
the  amount  of  $8|^ :  how  much  had  he  remaining  ? 

16.  A  drover  bought  4  cows  for  $168J,  and  after  paying 
$34^  for  pasturage,  he  sold  them  for  I203J-;  did  he  gain 
or  lose,  and  how  much  ? 

17.  A  man  traveled  in  a  certain  direction  34^  miles  the 
first  day,  and  37f  miles  the  second  day ;  then,  retracing  his 
path,  he  traveled  28^  7niles  on  the  third  day,  and  34^  miles 
on  the  fourth  day :  how  far  was  he  then  from  the  starting 
point  ? 

18.  What  is  the  difference  between  57J  yds.  +  72^^  yds. 
and  211  yds.  —  94^  yds.  9 

19.  A.  bought  a  house  for  $11,320^,  and  after  expending 
$1,31  If  for  repairs,  sold  it  for  $12,500 ;  did  he  gain  or  lose, 
and  how  much  ? 

MULTIPLICATION    OF    FRACTIONS. 
DEFINITION. 

78.  Multiplication  of  Fractions  is  the  operation 
of  finding  the  product  of  two  or  more  fractions. 

MENTAL     EXERCISES. 

1.  What  is  -J  of  5  ?  I  of  5  ?  I  of  5  ?  What  is  5  taken  ^ 
of  1  time  ?  5  taken  f  of  one  time  ?  5  taken  |  of  1  time  ? 
What  is  5  x  i?  5  x  |  ?  5  x  |?  5  x -f?  5  x  H? 


100  COMMOIT     FRACTIONS. 

2.  If  ^  of  an  orange  is  divided  into  5  equal  parts,  how 

much  of  a  whole  orange  is  1  of  these  parts  ?  how  much  is 

2  of  the  parts  ?   3  of  the  parts  ?    What  is  |  of  ^  ?  |  of  ^  ? 

fofi?    WhatisJ  xi?  ix  I?  ix|?  }xi? 

Explanation. — The  expression  ^  of  |  is  equivalent  to  the  ex- 
ession  I  x  I;  for  4  of  -^  is  the  sar 
therefore  equal  to  |^  x  |^  (Art.  27). 

3.  What  is  4-  of  i  ?  f  of  ■^?  f  of  ^  ?  What  is  ^  of  f  ? 
f  of  I  ?  What  is  4-  X  i?  f  X  i  ?  f  X  I  ?  I  X  f  ?  Is 
there  any  difference  between  f  of  f ,  and  f  of  ■§  ?  What 
isf  xl?    What  is  f  x|?  A  xf  ? 

OPERATION     OF     MULTIPLICATION. 

79.   Let  it  be  required  to  multiply  f  by  -f . 

Explanation. — We  first  multi-  opbbation. 

ply  f  by  5,  which,  according  to  Prin-         3       5        3x5        15 

ciple  1°  (Art.  65)  gives  ?-^;  but         ^       '^        4  X  7  ""  28 

this  result  is  7  times  the  required  product^  because  the  multiplier 
used  is  7  times  the  given  multiplier  ;  hence,  to  find  the  true  product, 
we  must  divide  it  by  7,  which,  according  to  Principle  3°  (Art.  65), 

3  X  5         15 

gives  -; =,  or  —r . 

^  4x7         28 

In  like  manner  we  may  treat  all  similar  cases ;  hence,  the  fol- 
lowing 

RULE. 

Reduce  the  factors  to  the  form  of  simple  frac- 
tions ;  then  multiply  the  num^erators  together  for  a 
new  numerator,  and  the  denominators  for  a  new 
denominator. 

E  X  AM  PLES.        3 

1.  Multiply  7i  by  |.  Ans.  ^  x  ^  =  |  =  4i. 

2.  Multiply  2i  by  ^  of  f  Ans.  ^  x  ^  x  ^  =  g- 


MULTIPLICATION".        >    '  '^  '    V   ,        foi 

Note.— After  indicating  the  operation,  m^wcJ^'oVciy^* factor* tKiit'it? 
common  to  any  numerator  and  any  denominator.  If  the  final  result 
is  an  improper  fraction,  reduce  it  to  a  whole,  or  to  a  mixed  number  ; 
if  it  is  a  proper  fraction,  reduce  it  to  its  lowest  terms. 

The  rule  may  be  simplified  in  the  following  cases : 
1°.  To  multiply  a  whole  number  by  a  simple  fraction  : 
Multiply  it  by  the  numerator  of  the  fraction  and 
divide  the  result  by  the  denominator. 

3.  Multiply  928  by  |.     Ans.  ?|?  x  |  =^^^2^  =  348. 

1         o  o 

2°.  To  multiply  a  whole  number  by  a  mixed  number : 
Multiply  it  first   by  the  fractional  part  of  the 

mixed  niunber,  then  by  the  integral  part,  and  find 

the  sum  of  the  results. 

(4.)     928  (5.)    3)1143 

6i  n 

348      I  of  928.  381      i  of  1143. 

5568      6  times  928.  8001       7  times  1143. 


5916      Product.  8382      Product. 

Perform  the  following  indicated  operations : 

6.  f  Xt%.  15.  f  of7ix}of90. 

7.  V  X  f  16.  345  X  4|. 

8.  1  X  12|.  17.  3f  X  2f. 

9.  7i  X  8i.  18.  I  X  I  X  A- 

10.  ^  X  61|.  19.  3i  X  1  X  llf 

11.  I  of  7  X  f.  20.  A  X  A  X  ^. 

12.  154  x^.  ^1-  3ix  WVx^. 

13.  6f  x  -W-  2^-  3i  X  7i  X  H- 

14.  ^  off  xi|.  23.  ^x2i  X  Hi 


i02f  ''     '■    '  '    'COMMON     FRACTIONS. 

^ '  '^^i^^D^rui^'k^^.  29.  A  of  3i  X  H. 

25.  2|  X  5|  X  iff.  30.  tt  X  f  X  3f . 

26.  114m  X  81^.  31.  2174-  X  112|. 

27.  f  ofA  xH.  32.  «x-Jx  A- 

28.  2i  X  If  X  |.  33.  8i  X  8J  X  8J. 

34.  (56|  +  24i)  X  (13i  +  9f )  =  ? 

35.  (Ill  -f-  302i)  X  (107^  -  30^)  =  ? 

36.  (207|  -f  39i)  X  (lOO^ij:  -  66^V)  =  ? 

37.  (4451  -  36i)  x  (36f  -  21^)  =  ? 

38.  (999f  -  ^^)   X  m^  -  72|)  =  ? 

39.  (256  -  7i)  X  (394  +  ^)  =? 

40.  (224  +  3A)  X  (88|  -4^)=? 

PRACTICAL    PROBLEMS. 

1.  If  a  man  earns  133^  per  week,  how  much  will  he  earn 
in  a  year  of  52  weeks  ?  A^is.   11,733^. 

2.  A  farmer  bought  43  acres  of  land  at  |104f  per  acre, 
16  cows  at  I28f  each,  and  2  plows  at  II 1^^  each ;  what 
did  they  all  cost  him  ?  Ans.   $4,985-^. 

3.  What  must  be  paid  for  600  barrels  of  flour  at  $5.37^ 
per  barrel  ?  Ans.  $3,225. 

4.  What  is  the  cost  of  33^  lbs.  of  tea  at  93f  cents  a 
pound?  Ans.   $31.25. 

5.  If  a  man  can  travel  7f  miles  in  1  hour,  how  many 
miles  can  he  travel  in  6J  hours  ? 

6.  If  it  takes  If  bushels  of  wheat  to  sow  an  acre,  how 
many  bushels  will  it  take  to  sow  7^^  acres  ? 

7.  A  grocer  bought  100  barrels  of  flour  at  $6-J  per  bar- 
rel ;  he  sold  49  barrels  at  17^  per  barrel,  and  the  rest  at 
$7^  per  barrel :  how  much  did  he  gain  ? 


MULTIPLICATION.  103 

8.  A.  bought  319f  acres  of  land  at  $200  per  acre ;  he 
then  sold  250f  acres  at  $250  per  acre,  and  the  remainder 
at  $26  6f  per  acre  :  how  much  did  he  gain  ? 

9.  A  drover  bought  64  sheep  at  $7f  a  piece  ;  he  then 
sold  30  of  them  at  $6|^  a  piece,  and  the  remainder  at 
f  8  J  a  piece :  did  he  gain  or  lose,  and  how  much  ? 

10.  A.  starts  from  Cincinnati  and  travels  at  the  rate  of 
5f  miles  an  hour ;  at  the  end  of  3|-  hours  B.  starts  from 
the  same  place  and  travels  in  pursuit  at  the  rate  of 
6 J  miles  an  hour :  how  far  apart  are  they  at  the  end  of 
5;^  hours  ? 

11.  What  is  the  cost  of  f  of  a  piece  of  cloth  containing 
13J  yards  at  $2J  a  yard  ? 

12.  A.,  B.,  and  C.  own  a  tract  of  land;  A's  share  is 
62|-  acres,  B's  share  is  1^  times  as  much  as  A's,  and 
C's  share  is  lOJ  acres  greater  than  A's  and  B's  together : 
how  many  acres  in  the  whole  tract  ? 

13.  A  man  traveled  112J  miles  in  3  days ;  the  first  day 
he  traveled  f  of  the  whole  distance,  and  the  second  day  he 
traveled  ^  of  the  distance  he  did  the  first  day :  how  far  did 
he  travel  the  third  day  ? 

14.  A  woman  is  24f  years  old,  and  her  husband  lacks 
7f  years  of  being  twice  as  old ;  what  are  the  united  ages 
of  the  two  ? 

15.  What  will  f  of  -J  of  a  yard  of  cloth  cost  at  the  rate 
of  A  of  $3|  per  yard? 

16.  How  many  yards  in  8  pieces  of  cloth,  each  contain- 
ing 37|  yards  ? 

17.  If  a  train  of  cars  runs  22j^  mi,  an  hour,  how  far  will 
it  run  in  8 J  hrs  9 


104  COMMON"     FE  ACTIONS. 

DIVISION    OF    FRACTIONS. 
DEFINITION. 

80.  Division  of  Fractions  is  the  operation  of  find- 
ing the  quotient  of  one  fraction  by  another. 

MENTAL     EXERCISES. 

1.  In  1  orange,  how  many  ffths  of  an  orange  9  How 
many  times  is  \  contained  in  1  ?  What  is  the  quotient  of 
Ibyi?  oflby^?  of  1  by  |  ? 

2.  What  is  the  quotient  of  1  by  I  ?  of  2  by  |  ?  of  3  by 
\  ?  How  do  you  divide  an  integral  number  by  a  fractional 
unit? 

Explanation.— The  quotient  of  1  by  |  is  7  ;  but  the  quotient  of 

3  by  \  is  3  times  as  great  as  tlie  quotient  of  1  by  | ;  bence  it  is  3  x  7, 
that  is,  we  multijply  the  given  nwriber  ly  the  denominator  of  the  frac- 
tional unit. 

3.  What  IS  the  quotient  of  3  by  ^  ?  of  |  by  ^^  ?  of  }  by 

\  ?    How  do  you  divide  a  simple  fraction  by  a  fractional 

unit  ? 

Explanation. — The  quotient  of  3  by  |  is  3  x  7 ;  but  the  quotient 
of  f  by  I  is  only  \  as  great  as  the  quotient  of  3  by  \  ;  hence  it  is 

3x7 

— 2— ,  that  is,  we  multiply  the  numerator  of  the  given  fraction  by  the 

denominator  of  the  fractional  unit. 

4.  What  is  the  quotient  of  f  by  1  ?  of  |  by  2  ?  of  f  by 
5  ?  How  do  you  divide  a  simple  fraction  by  a  whole  num- 
ber ? 

Explanation.— The  quotient  of  f  by  lis  | ;  but  the  quotient  of 
f  by  5  is  only  \  as  great  as  the  quotient  of  f  by  1 ;   hence  it  is 

SI  S 

;  X  =    or    - — -,  that  is,  we  multi'ply  the  denominator  of  the  given 

4  5  4x5 

fraction  by  the  whole  number. 


DIVISION.  105 

OPERATION     OF     DIVISION. 

81.  Let  it  be  required  to  divide  -f-  by  ^ : 

Explanation. — Here  the  divi-  operation. 

sor  is  equal  to  i  taken  4  times,  that  3        4        3/1  \ 

is,  to  1^  X  4  ;  hence,  to  find  the  quo-  -;z  -^  -^  =  -^  -r-  (^  X  41 
tient  we  divide    f    by  ^  and  that  ^  ' 

result  by  4.      To   divide  f    by  i  3x5        3        5        15 

we  multiply  its   numerator  by  5  = =  -  x  -  =  — ■» 

3x5  7x474       28 

(Art.  80,  Ex.  3),  which  gives  -y-  ; 

to  divide  the  result  by  4  we  multiply  its  denominator  by  4,  (Art.  80, 

Ex.  3),  which  gives  = — ^,  and  this  is  the  same  thing  as  =  x  2,  or  — . 

Here  we  have  inverted  the  divisor,  that  is,  we  have  made  its  terms 
change  places,  and  then  we  have  proceeded  as  in  multiplication. 
In  like  manner  we  may  treat  all  similar  cases  ;  hence,  the 

RULE. 
Reduce  both  dividend  and  divisor  to  simple  frac- 
tions;    then  invert  the  divisor  and  proceed  as  in 
multiplication. 

Note. — Before  performing  the  multiplication  cancel  and  reduce 
as  explained  in  Art.  79. 

EXAMPLES. 

1.  Divide  Z\  by  |.  Ans.  ^  x  |  =  y  =  5^. 

To  divide  a  whole  number  by  a  simple  fraction  we  may 
Multiply  it  hy  the  denom^inator  of  the  fraction  and 
divide  the  result  hy  the  numerator. 

2.  Divide  27  by  f .        Ans.  y  x  |  =  ^^-  =  331- 
Perform  the  following  indicated  operations  : 


106 


C0MM02S'     rRACTIONS. 


5.  *^«. 

21.  fofJI 

-AofH. 

6.  tt  -  A- 

22.  1  of  7|- 4  of  2,15. 

7.  iM-«- 

•     23.  (8i  +  3i)  -  7|. 

8.  241  -r-  ^. 

24.  (7f +  8i)H-iof6^. 

9.  1275  -J-  ff. 

25.  iof4| --(2^+31). 

10.  f  -  t  X  A- 

26.  7f  X  84-T-3i  X  ^. 

11.  V-^lof-A. 

27.  ^ofSJ-^aiJ  X  7i. 

12.  iofif-v-if  XtV- 

28.  (3+  +  15f )  -  27A- 

13.  |of«-HofJ. 

29.  25^  -^ 

-  m  +  5i). 

14.  W- 

-fof*. 

30.  llT»r  - 

-  (^  +  ^T^r)- 

15.  W  - 

-¥• 

81.  14A  - 

-  f  of  15. 

16.  ^^- 

r21.      • 

32.  214}  - 

hiof25ii. 

17.  m- 

-i^. 

33.  i  of}  of  5 -=-21^. 

18-  54  -=-  Sf. 

34.  4  X  7i  -=-  8  X  19I-. 

19.  611^  H-  20A- 

35.  i  of  15|  -=-  9i  X  f 

20.  10015 

\  -  66f. 

36.  (32i  +  7})H-f  of-A. 

37.  2J  X  3f  +  3  -^  4^  -  7|:  =  ? 

38.  (3^  +  7f )  --  2A  +  71-  ^  5f  =  ? 

39.  (Hi  +  17})  --  (33i  _  4J)  +  8^  zz:  ?       • 

40.  (i  of  6}  -  24)  -^  (25  +  1^  of  3|)  =  ? 

Note. — The  following  problems  afford  exercises  on  all  the  opera, 
tions  that  can  be  performed  upon  fractions  : 

PRACTICAL     PROBLEMS. 

1.  If  Hi  yds.  of  silk  cost  $13,  what  is  the  cost  of  1  yard? 
of  5  yds.  ?  Ans.  UH;  and  I8|. 

2.  If  37|  ounces  of  silver  cost  $31J,  what  is  the  cost  of 
1  ounce?  of  150  ounces  ?  Ans.  If;  and  $125. 

3.  If  3|  hu.  of  buckwheat  cost  $2|,  what  does  1  bu.  cost  ? 
What  is  the  cost  of  30  bu. ?  A71S.  %\^\  and  119. 

4.  A.  divides  $3,000^  into  7  equal  shares  and  gives  ij 


DIVISIOIT.  107 

of  these  shares  to  a  benevolent  society ;  how  much  does 
he  give  to  the  society?  Ans.  $1,928||. 

5.  A.  can  build  a  wall  in  10  days,  B.  can  do  it  in  12 
days,  and  0.  in  15  days ;  what  part  of  the  wall  can  they 
all  build  in  1  day  ?  A^is.  xd"  +  tV  +  tt  —  i- 

6.  How  long  will  it  take  them  all  to  build  the  wall  ? 

Explanation. — Because  it  takes  1  day  to  build  \  of  the  wall,  it 
will  take  4  days  to  build  the  whole. 

7.  What  number  multiplied  by  1|  will  give  14}  ? 

8.  The  difference  of  two  numbers  is  15-^  and  the 
greater  number  is  20}^ ;  what  is  the  less  number  ? 

9.  A  man  inherits  f  of  an  estate  and  gives  his  son  -J-  of 
his  share ;  what  part  of  the  estate  does  the  son  receive  ? 

10.  If  f  of  a  ton  of  coal  costs  $13,  what  will  7  tons  cost  ? 

11.  If  coffee  costs  13 J  cents  a  pound,  how  much  can  be 
bought  for  $10  ?  for  $16  ? 

12.  How  many  pounds  of  coffee  can  be  bought  for  1784 
at  $^  per  pound  ?  at  $|  ? 

13.  A.  can  do  a  piece  of  work  in  3  days  and  B.  can  do 
it  in  2  days;  liow  long  will  it  take  them  both  to  do  it? 

14.  A.  bought  24:^  yds.  of  cloth  at  $4^  a  yard,  and  sold 
the  whole  for  $1 28| ;  what  did  he  gain  per  yard  ? 

15.  How  many  pounds  of  sugar  at  12J  cts.  a  pound 
must  be  given  for  16-|  lbs.  of  butter  at  22^  cts.  a  pound  ? 

16.  If  6  men  can  do  a  piece  of  work  in  7^  days,  how  long 
will  it  take  one  man  to  do  it  ? 

17.  If  a  man  can  walk  10^  miles  in  IJ  hours,  how  far 
can  he  walk  in  1  hour  ?  in  5|-  hours  ? 

18.  A  merchant  owning  ^^  of  a  vessel,  sold  |  of  his  share 
for  $1,640 ;  what  was  the  vessel  worth  at  that  rate  ? 


108  COMMON     FRACTIONS. 

19.  A.  can  mow  a  piece  of  grass  in  4  days,  and  B.  can 
do  it  in  2  days ;  how  long  will  it  take  both  to  do  it  ? 

20.  A  man  made  a  journey  in  6J  days,  traveling  at  the 
rate  of  22|  miles  a  day ;  on  his  return  he  traveled  at  the 
rate  of  24|-  miles  a  day :  how  many  days  did  it  take  him  to 
return  ? 

21.  A  merchant  bought  a  piece  of  cloth  containing 
SQ^ycls.  for  $G5J ;  at  what  rate  must  he  sell  it  per  yard  so 
as  to  gain  125^  ? 

22.  A.  sets  out  from  Detroit  and  travels  towards  Buffalo 
at  the  rate  of  6f  miles  an  hour;  at  the  end  of  2 J  hours  B. 
sets  out  from  Detroit  and  follows  at  the  rate  of  8;^  miles 
an  hour:  how  far  apart  are  they  at  the  end  of  5f  hours? 

23.  A  farmer  sold  to  a  grocer  32^  bu.  of  com  at  ^^  a 
bushel,  and  86  lbs.  of  butter  at  If  a  pound.  He  received 
in  pay  200  lbs.  of  sugar  at  $^  a  pound,  and  the  remainder 
in  money  ;  how  much  money  did  he  receive  ? 

24.  A  regiment  lost  220  men  in  battle,  which  was  4  men 
more  than  f  of  the  whole  regiment ;  how  many  men  were 
there  in  the  regiment  ? 

25.  A.  owned  J  of  a  ship  and  sold  f  of  his  share  to  B. ; 
B.  then  sold  f  of  what  he  bought  to  C.  for  $3000;  what 
was  the  whole  ship  worth  at  that  rate  ? 

CONTRACTIONS    IN    MULTIPLICATION    AND    DIVISION. 

83.  The  rules  for  multiplication  and  division  of  frac- 
tions lead  to  certain  contractions  in  multiplication  and 
division  of  whole  numbers,  of  wliich  the  following  are 
some  of  the  most  important. 

1°.  The  fraction  J-f^  is  equal  to  25  ;  hence,  to  multiply  a 
number  by  25,  we  may 


DIVISION.  109 

Annex  2  ciphers  and  divide  the  result  by  4- 

To  divide  a  number  by  25,  we  may 

Multiply  it  by  4  cu^^^  divide  the  result  by  100. 

EXAM  PLES. 

1.  Multiply  3,416  by  25.  Ans.  ?i^  =  85400. 

2.  Divide  5,875  by  25.  Ans.  ^^^^^  ^  =  235. 

3.  394  X  25  =  ?  8.  9,850  ^  25  =  ? 

4.  3,724  X  25  =  ?  9.  93,100  ^  25  =  ? 

5.  8,123  X  25  =  ?  10.  87,525  -^  25  =  ? 

6.  10,201  X  25  =  ?  11.  46,350  -^  25  =  ? 

7.  4,386  X  25  =  ?  12.  174,025  ^  25  =  ? 
Let  the  pupil  deduce  rules  for  multiplying  and  dividing 

by  12J,  by33i,  andbyl25: 

13.  81  X  12i  =  ?  21.  10,125  -^12^=? 

14.  914  X  124  =  ?  22.  $11,425  -^  $12^  =  ? 

15.  $4,834  X  12i  =  ?  23.  9,125  yds.  -M2i  =  ? 

16.  1375  X  331  ^  ?  24.  13,500  ^33-^=? 

17.  28,452/^5.  X  33^  =  ?  25.  $29,100  -^334=? 

18.  l,%tQyds.   X  125  =  ?  26.  8,700  «/^5.-^33i?/^5.  =  ? 

19.  $4,365  X  125  =  ?  27.  $2,250  -^  125  =  ? 

20.  34,115  ijds.  X  124  =  ?  28.  10,000/5.  -j-  125  =  ? 

REVIEW     QUESTIONS. 

(60.)  What  is  a  fractional  unit?  What  is  a  half,  a  third,  a 
fourth,  &c.  ?  What  is  the  reciprocal  of  a  number  ?  (61.)  What  is 
a  fraction  ?  How  do  you  write  a  common  fraction  ?  What  is  the 
denominator?  the  numerator?  What  are  terms?  (63.)  In  how 
many  ways  may  we  regard  a  fraction  ?  Illustrate.  (64.)  What  is 
a  proper  fraction?    Illustrate.     An  improper  fraction?    Illustrate. 


110  DECIMAL     FRACTIONS. 

A  mixed  number  ?  Illustrate.  A  simple  fraction  ?  Illustrate.  A 
complex  fraction  ?  Illustrate.  (65.)  State  the  fundamental  prin- 
ciples of  fractions.  (66.)  What  is  reduction?  (67.)  Give  the  rule 
for  reducing  a  whole  number  to  a  fraction  with  a  given  unit. 
(68.)  Give  the  rule  for  reducing  a  mixed  number  to  a  fraction. 
(69.)  Give  the  rule  for  reducing  an  improper  fraction  to  a  mixed 
number.  (70.)  Give  the  rule  for  reducing  a  fraction  to  its  lowest 
terms.  (7ii.)  Give  the  rule  for  reducing  fractions  to  a  common 
denominator.  (73.)  Give  the  rule  for  reducing  fractions  to  their 
least  common  denominator.  (74.)  What  is  addition  of  fractions? 
(75.)  Give  the  rule  for  addition  of  fractions.  (76.)  What  is  sub- 
traction of  fractions  ?  (77.)  Give  the  rule  for  subtraction  of  frac- 
tions. (78.)  What  is  multiplication  of  fractions  ?  (71).)  Give  the 
rule  for  the  multiplication  of  fractions.  (80.)  What  is  division  of 
fractions?  (81.)  Give  the  rule  for  division  of  fractions.  (82.) 
Give  a  rule  for  multiplying  by  25.     Give  a  rule  for  dividing  by  25. 


II.      DECIMAL     FRACTIONS. 

DEFINITIONS. 

83.  A  Decimal  Fraction  is  a  fraction  whose  denom- 
inator is  10,  100,  1,000,  or  some  higher  power  of  10,  (Art. 
46),     Thus,  ^,  -jig^,  yfoir?  ^^c,  are  decimal  fractions. 

MENTAL      EXERCISES. 

1.  If  1  is  divided  into  10  equal  parts,  what  is  one  of  the 
parts  called  ?    2  of  the  parts  ?     5  of  the  parts  ? 

2.  If  i^g-  is  divided  into  10  equal  parts,  what  is  one  of 
the  parts  called  ?    2  of  the  parts  ?    17  of  the  parts  ? 

8.  If  yJ^  is  divided  into  10  equal  parts,  what  is  1  of  the 
parts  called  ?    3  of  the  parts  ?    27  of  the  parts  ? 

4.  WhatisT^oofTV?  Aof-rb?  A  of  1,000  ?  -^  of 
10,000?  What  power  of  10  is  100?  1,000?  10,000? 
100,000?     1,000,000? 


REDUCTION.  Ill 

DECIMALS,      AND     THE     DECIMAL     POINT. 

84.  Decimal  fractions  may  be  written  in  two  ways: 
their  denominators  may  be  expressed,  as  in  ordinary 
fractions;  or  their  denominators  may  be  indicated 
by  means  of  a  point  followed  by  one  or  more  figures.  In 
the  latter  case  they  are  called  Decimals,  and  the  point  (.) 
used  in  writing  them  is  called  the  Decimal  Point. 

NOTATION      OF     DECIMALS. 

S5.  Decimals  are  written  in  the  same  manner  as  whole 
numbers,  and  both  may  be  written  together,  decimals  on 
the  right  and  whole  numbers  on  the  left,  as  shown  in  the 
following 

NUMERATION     TABLE. 


^  o  I  ^  's  I  ^    .  ^.    4  I  I 

Oil— irta53?HcoJ2  rSS?5 


^ 


Si 


3963042.5749826.. 


v_ 


Whole  Numbers.  Decimals. 

Note. — In  whole  numbers,  places  of  figures  and  orders  of  units 
are  counted  from  the  decimal  point  toward  the  left ;  in  decimals,  they 
are  counted  from  the  decimal  point  toward  the  right. 

A  figure  in  the  first  place  of  decimals  denotes  tenths ; 
in  the  second  place  it  denotes  hundredths ;  in  the 
third  place  it  denotes  thousandths ;  and  so  on,  as 
indicated  in  the  table.  Hence,  to  write  a  decimal  we  have 
the  following 


113  DECIMAL    FKACTIONS. 

RULE, 

Write  the  number  of  tenths  in  the  first  decimat 
place,  the  number  of  hundredths  in  the  second  place, 
the  number  of  thousandths  in  the  third  place,  and 
so  on. 

EXAM  PLES. 

1.  Write  three fte7iths,  as  a  decimal.  Ans.  .3. 

2.  Write  twenty  seven/ hu7idredths.  Ans.  .27. 

3.  ^YvitG  forty  eight /thousandths.  Ans.  .048. 

Note. — In  Example  2,  because  27  hundredths  is  the  same  as  2 
tenths  and  7  hundredths,  we  write  2  in  tlie  first  place  of  decimals 
and  7  in  the  second  place.  In  Example  3,  because  48  thousandths  is 
the  same  as  0  tenths,  4  hundredths,  and  8  thousandths,  we  write  0  in 
the  first  place,  4  in  the  second  place,  and  8  in  the  third  place.  Let 
the  student  in  like  manner  explain  each  of  the  following  examples : 

4.  Two  hundred  mid  thirteen /thousandths. 

5.  One  thousand  and  six/ten  thousandths. 

6.  Four  thousand  two  hundred  and  seven/millionths. 

7.  Two  hundred  and  seventy  four  thousand  three  hun- 
dred and  forty  three /millionths. 

8.  Twenty  three  million,  two  hundred  and  four  thousand, 
five  hundred  and  seventy  seven/hundred  milUonths. 

A  mixed  decimal  is  a  mixed  number  whose  fractional 
part  is  a  decimal.  Thus,  six,  and  three/ tenths  is  a  mixed 
decimal;  it  may  be  written  6.3.  In  all  such  cases  the 
integral  part  is  written  on  the  left  of  the  decimal  point. 

9.  Twenty,  2Lndi  forty  four /hundredths.      Ans.  20.44. 

10.  Thirty  seven,  and  seventy  two /thousandths. 

11.  Forty  seven,  and  two  hundred  and  nine /milUonths. 
Note. — In  the  preceding  examples,  decimals  are  in  italics.    The 

sign  /  separates  the  numerator  from  the  denominator. 


REDUCTION".  113 

From  what  precedes,  we  see  that  a  decimal  fraction  may 
Be  expressed  decimally  by  writing  its  numerator,  and 
then  placing  a  decimal  point  so  that  the  number  of  figures 
following  it  shall  be  equal  to  the  number  of  ciphers  in  the 
denominator.  If  the  number  of  figures  in  the  numerator 
is  less  than  the  number  of  ciphers  in  the  denominator,  a 
sufficient  number  of  ciphers  must  be  prefixed,  that  is, 
tvritten  before  the  numerator. 

EXAMPLES. 

1.  A  =  -3.  4.  9tWV  ::=  9.313. 

2.  Tff^  =  .045.  5.  4r«o  =  4.079. 

3.  tMt^  =  .0017.  6.  256TnjW^  =  256.00117. 

NUMERATION     OF     DECIMALS. 

86.  From  what  has  been  explained,  we  see  that  a  deci- 
mal may  be  read  by  the  following 

RULE. 

Bead  the  significant  part  as  a  whole  number,  and 

add  the  name  of  the  lowest  unit  of  the  decimal. 

Note. — Before  reading  a  decimal  the  pupil  should  numerate  it, 
that  is,  he  should  begin  at  the  left  hand  and  name  the  units  of  each 
place  :  thus,  tenths,  hundredths,  thovsandths,  etc. ,  according  to  the 
table. 

Read  the  following  decimals : 

1.  .087.  Ans.  Eighty  seven j thousandths. 

2.  .000317. 

Ans.  Three  hundred  and  seventeen  I  millionths. 

3.  .0027.  6.  .52346.  9.  .11122. 

4.  .10364.  7.  .50067.  10.  .224785. 

5.  .00201.  8.  .320315.  11.  .0067412. 


114  DECIMAL     FRACTIONS. 

Note. — In  mixed  decimals  we  read  the  integral  and  the  decimal 
parts  separately. 

12.  120.009. 

Ans.  One  hundred  and  twenty,  and  nine/thousandths. 

13.  19.00015.  15.  150.15632.  17.  45.36251. 

14.  212.1236.  16.  34.001725.  18.  111.009265. 

DECIMAL     CU  RRENCY. 

87.  The  currency  of  the  United  States  is  purely  deci- 
mal, the  primary  unit  being  1  Dollar.  In  it  Dimes  are 
te?iths  of  a  dollar,  Cents  are  hundredths  of  a  dollar,  and 
Mills  are  thousandths  of  a  dollar.  Dollars,  cents,  and 
mills  are  generally  written  in  the  form  of  a  mixed  decimal, 
the  decimal  point  being  placed  after  dollars.  Thus,  the 
expression  174.853,  denotes  74  dollars,  8  dimesy  5  cents,  and 
3  mills  ;  it  is  read  75  dollars  85-]^  cents. 

Note. — An  eagle  is  equal  to  $10.  In  business  transactions  the 
terms  eagle,  dime,  and  mill  are  but  little  used,  sums  of  money  being 
expressed  in  dollars  and  cents. 

FUNDAMENTAL    PRINCIPLES. 

88.  Moving  the  decimal  point  one  place  to  the  right 
changes  tenths  to  units,  hundredths  to  tenths,  and  so  on ; 
but  this  is  equivalent  to  multiplying  the  decimal  by  10 : 
hence,  the  following  principle : 

1°.  Moving  the  decimal  point  one  place  to  the  right  is 
equivalent  to  multiplying  the  decimal  hy  10. 

In  like  manner  we  have  the  following  principle : 

2°.  Moving  the  decimal  point  one  place  to  the  left  is 
equivalent  to  dividing  the  decimal  by  10. 

Annexing  a  cipher  to  a  decimal  multiplies  both  numera- 
tor and  denominator  by  10 ;  but  this  does  not  alter  the 


REDUCTIOJS^.  115 

value   of  the  fraction   (Art.  65);   hence,  the  following- 
principle  : 

3°.  Annexing  one  or  more  ciphers  to  a  decimal  does  not 
change  its  value. 
In  like  manner  we  have  the  following  principle  : 
4°.  Striking  out  one  or  more  terminal  ciphers  does  not 
change  the  value  of  a  decimal, 

REDUCTION    OF    COMMON    FRACTIONS    TO    DECIMALS. 

89.   Let  it  be  required  to  reduce,  that  is,  to  change  I  to 
the  form  of  a  decimal. 

Explanation. — The  value  of  |  is  equal  to  opbratiox. 

5  -J-  8  (Art.  &2) ;  to  find  this  quotient  we  annex  8"\'=i00n 

three  ciphers  to  5,  which  is  equivalent  to  mul- 

tiplying   it  by   1,000,   and  then    perform  the  .625 

division  ;  but  this  result  is  1,000  times  as  great 
as  the  true  value  of  the  fraction  ;  we  therefore  divide  it  by  1,000, 
which   is  done  by  pointing    oflf   three    decimal  figures   (Priii.    2, 
Art.  88). 

In  like  manner  we  may  treat  all  similar  cases ;  hence,  the 

RULE. 

Annex  ciphers  to  the  numerator  and  divide  the 

result  by  the  denominator ;  then  point  off  from  the 

right  of  the  quotient  a  number  of  decimal  figures 

equal  to  the  number  of  ciphers  annexed. 

Note. — If  the  number  of  figures  in  the  quotient  in  less  than  the 
number  of  ciphers  annexed,  prefix  the  requisite  number  of  ciphers. 

EXAMPLES. 

Reduce  the  following  fractions  to  decimals : 

3.  fj.  6.  ^^.  9.  r^. 


116  DECIMAL     FEACTIONS. 

Note. — To  reduce  a  mixed  number  to  a  decimal  form,  we  reduce 
the  fractional  part  to  a  decimal  and  annex  the  result  to  the  inte- 
gral part. 


10.  19J. 

13.  11-^. 

14-  ai^Ar- 

11. 2m. 

13.  110^17. 

15-*A%- 

APPROXIMATE     RESULTS. 

90,  Ifc  may  happen  tliat  the  division  described  in  the 
last  article  will  not  terminate,  no  matter  how  many  ciphers 
we  annex.  In  this  case  the  decimal  found  by  stopping 
at  any  particular  step  of  the  division  is  called  an  approx- 
imate value  of  the  given  fraction.  Thus  .1904  is  an 
approximate  value  of  ^.  'In  this  case  ^  is  greater  than 
.1904  and  less  than  .1905  ;  hence,  it  differs  from  either  by 
less  than  they  differ  from  each  other,  that  is,  by  less  than 
.0001.  In  like  manner  the  approximate  value  of  a  frac- 
tion found  ,by  stopping  at  any  decimal  figure  differs  from 
the  true  value  of  the  fraction  by  less  than  the  correspond- 
ing decimal  unit. 

If  we  stop  at  any  decimal  figure  and  increase  it  by  1 
when  the  next  figure  is  equal  to,  or  greater  than  5,  the 
error  can  never  exceed  ^  the  corresponding  decimal  unit. 
Thus,  f  =  .667,  and  -J-  =  .333,  each  to  within  less  than  -J 
of  .001.  This  is  the  practical  method  of  finding 
approximate  values  of  decimals. 

Note. — In  applying  the  principles  of  decimals  to  practical  cases 
we  shall  habitually  follow  the  method  of  approximation  just  ex- 
plained, and,  except  in  special  cases,  we  shall  limit  the  approximation 
either  to  three  or  to  four  decimal  places. 

In  United  States  money  we  shall  habitually  limit  the  approxima- 
tion to  three  decimal  places  ;  each  result  will  then  be  true  to  within 
Iialf  a  mill,  or  the  twentieth  of  a  cent. 


ADDITIOI^".  117 

EXAM  PLES. 

Reduce  the  following  fractions  to  decimals,  carrying  the 
approximation  to  the  fourth  place : 

1.  foff  6.  4f  f  11.  21|i. 

2.  «.  7.  iofSJ.  12.1^^. 

3.  lA.  8.  Aof^i  13.  14^. 

4.  A.  9.  f  of  I  of  44.  14.  IS^Vt- 

5.  dii.  10.  ^  X  4f.  15.  4fi|. 

ADDITION    OF    DECIMALS. 
DEFINITION. 

91.  Addition  of  Decimals  is  the  operation  of  find- 
ing the  sum  of  two  or  more  decimals. 

MENTAL      EXERCISES. 

1.  AVhat  is  the  sum  of  4  tenths  and  5  tenths?  of  .3 
and  .6  ?  How  many  units  and  tefiths  of  a  iinvt  in  the  sum 
of  .8  and  .9  ?  What  is  the  sum  of  .3,  .5,  and  .9  ?  of  .5, 
.7,  .9,  and  .6  ? 

2.  How  would  you  read  49  hundredths  in  tenths  and 
hundredths  9  How  many  hutidredfhs  are  there  in  6  te?iths? 
What  is  the  sum  of  .5  and  .49  ?  of  .59  and  .4  ?  How 
would  you  read  three  hundred  and  seventeen/thousandths 
in  tenths,  hmidredfhs,  and  thousandths  ? 

3.  What  is  the  sum  of  13  cts.  and  22  cts.  ?  of  $.13  and 

1.22  ?     Is  there  any  difference  between  13  cts.  and  $.13  ? 

What  is  the  sum  of  25  cts.  and  $.45  ?  of  $.4,  1.25,  and  1.7  ? 

of  .4,  .25,  and  .7  ? 

Note. — Addition  of  decimals  depends  on  tlie  same  principles  as 
addition  of  integers. 


118  DECIMAL     FRACTIOlfS. 

OPERATION    OF    ADDITION. 

93.  Let  it  be  required  to  find  the  sum  of  4.035,  76.19, 
and  114.0305. 

Explanation. — We  write  the  decimals  operation. 

so  that  units  of  the  same  order  shall  stand  in  4.035 

the  same  column  ;   this  will  bring  all  the  lyo  iq 

decimal  points  in  one  column:  then  begin-  /o.ly 

ning  at  the  right,  we  add  each  column  sep-  114.0305 

arately,  setting  doicn  and  carrying  as  in  sim-  ~ 

pie  numbers.    Hence,  the  Sum.     194.2555 

RULE. 
Write  the  decijnals  so  that  units  of  the  same  order 
shall  stand  in  the  same  column,  and  add  as  in  sim- 
ple numbers. 

Note. — The  decimal  points  of  the  numbers  to  be  added,  and  of 
their  sum,  must  stand  in  the  same  column. 


EXAMPLES. 

(1.) 

(2.) 

(3.) 

(4.) 

(5.) 

3.057 

5.6000 

5.43 

0.105 

$3.97 

14.086 

17.0032 

12.998 

0.0012 

$4,295 

209.3154 

35.9070 

317.0971 

0.25 

$11,464 

226.4584    58.5102    335.5251   0.3562   $19,729 

6.  Find  the  sum  of  .632,  .718,  3.202,  and  111.1. 

7.  Of  .0049,  47.0426,  37.041,  and  360.0039. 

8.  Of  $81,053,  $67,412,  $93,172,  and  $14.38. 

9.  Of  $59,317,  $69,565,  $8,213,  and  $7,775. 

10.  Of  3.25  Ihs.,  47.348  Ihs.,  748.4  lbs.,  and  29.32  lbs. 

11.  Of  672.5  yds.,  4.923  yds.,  80  yds.,  and  .0764  yds. 

12.  Of  72.5  +  140  +  340.03  +  21.5715  +  4.0008. 

13.  Of  2.8146  +  .0938  +  8.875  +  231.2788  +  4.0087. 


ADDITIOlf.  119 

14  Of  54.3  fU  +  7.29  ft  +  180.0046  ft  +  187  ft  + 
3.024 /if. 

15.  Of  57.038  +  95.00487  +  53.4690  +  107.00003. 

16.  Of  $62.70  +  $2.03  +  $4,009  +  $78.15  +  $114. 

17.  Of  .0009  +  3.0021  +  .128  +  8.0469  +  59. 

18.  Of  3.0102  lu,  +  11.5008  hu,  +  73.07  lu,  +  2.92  lu. 
H-  9.5  hu. 

19.  2.005  +  110.301  +  .069  +  7.375  +  2.25  =  ? 

20.  17.215  +  3.0567  +  2.072  +  4.009  +  54.75  =  ? 

21.  29.157  ft  +  8.0016  ft  +  77.29  ft  +  32.004/^.  + 
8.848 /f.  =  ? 

22.  14.2351  +  651.012  +  2.219  +  3.157  +  13.614  =  ? 

23.  $861.55  +  $378.25  +  $461.37  +  $683.57  +  $1,205.47  =  ? 

24.  213.7  U.  +  2.913  lu.  +  14.769  lu,  +  .0078  lu.  =  ? 

25.  15.753^^5.  +  2.069yds.  +  ItQUdyds.  +  10.27yds. 
+  3.2107  yds.  =  ? 

PRACTICAL    PROBLEMS. 

1.  A  boy  paid  28  cts.  for  a  slate,  75  cts.  for  paper,  and 
94  cts.  for  an  Arithmetic ;  what  did  he  pay  for  all  ? 

Ans. .  $1.97. 

2.  One  field  contains  5.3  acres,  a  second  contains 
11.43  acres,  a  third  contains  17.59  acres,  and  a  fourth 
contains  3.175  acres ;  how  many  acres  in  all  of  them  ? 

3.  A.  bought  16  hams  for  $31.87|^,  a  bag  of  coffee  for 
$17.92,  a  chest  of  tea  for  $12.75,  and  a  firkin  of  butter  for 
$21.37-|- ;  what  did  they  all  cost  ? 

4.  A  man  bought  candles  for  $6.89,  flour  for  $25.56, 
raisins  for  $1.12J-  (^.  e.  for  $1,125),  cheese  for  $8.37^-,  and 
sugar  for  $5.44 ;  what  was  the  cost  of  the  whole  ? 


1 20  DECIMAL     FRACTIOI^S. 

5.  A  merchant  sold  4-  pieces  of  muslin ;  the  first  con- 
tains 34.25  yds.f  the  second  38.056  yds.,  the  third  40.2  yds., 
and  the  fourth  37,225  yds. ;  how  many  yards  in  all  ? 

6.  A  farmer  has  4  bins  of  wheat ;  in  the  first  there  are 
86.35  hu.,  in  the  second  73.125  hu.,  in  the  third  96.5  hi , 
and  in  the  fourth  74.3  bu. ;  how  many  bushels  in  all  ? 

7.  In  5  piles  of  wood  there  are  respectively  4.316  cords, 
8.23  cords,  11.25  cords,  7.364  cords,  and  13.819  cords  ;  how 
many  cords  are  there  in  all  the  piles  ? 

8.  B.  bought  a  house  for  15,000,  a  store  for  $6,290,  mer- 
chandise for  123,654.12,  a  horse  for  1278.53,  a  farm  for 
19,371.60,  bank  stock  for  $11,500,  and  a  watch  for  892.72^; 
Avhat  did  the  whole  cost  him  ? 

SUBTRACTION     OF     DECIMALS. 

DEFINITION. 

93.  Subtraction  of  Decimals  is  the  operation  of 
finding  the  difference  between  two  decimals. 

MENTAL     EXERCISES. 

If  6  teyiths  are  taken  from  9  tenths,  how  much  will 
remain  ?  What  is  the  difference  between  23  tenths  and 
14  tenths  ?  23  tenths  is  equal  to  how  many  units,  and  how 
mmy  tenths? 

2.  What  is  the  difference  between  35  hundredths  and 
2  tenths  9  How  many  tenths  in  35  hundredths  9  What  is 
the  difference  between  .45  and  .25  ?  4.5  and  .6  ?  .7  and  .15  ? 
4.5  and  2.2  ?   3.5  and  2.7  ?   3.2  and  1.9  ? 

Note. — Subtraction  of  decimals  depends  on  the  same  principles 
as  subtraction  of  integers. 


SUBTRACTION.  121 

OPERATION     OF    SUBTRACTION. 

94.   Let  it  be  required  to  subtract  4.079  from  11.362. 

Explanation.  —  The    subtrahend    is  opbration. 

written  under  the  minuend,  so  that  units        Minuend  11  362 


Subtrahend,        4.079 


of  the  same  order  shall  stand  in  the  same 

column  ;  this  will  bring  the  decimal  points 

into  the  same  column:    the  operation  is        Remainder,        7.283 

then  performed  as  in  the  subtraction  of 

simple  numbers.     Hence,  the  following 

RULE. 

Write  the  subtrahend  under  the  minuend,  so  that 
units  of  the  same  order  shall  stand  in  the  saine 
column ;  then  subtract  as  in  simple  numbers. 

Note. — The  decimal  points  of  the  minuend,  of  the  subtrahend, 
and  of  the  remainder  must  stand  in  the  same  column. 


EXAMPLES. 

(1-) 

(2.) 

(3.) 

(4.) 

Minuend,        5.316 

17.0091 

1075.0567 

$312,475 

Subtrahend,    2.013 

11.9902 

287.9374 

$214,268 

Remainder,     3.303 

5.0189 

787.1193 

198.207 

Note. — If  the  subtrahend  contains  more  decimal  figures  than  the 
minuend,  annex  the  requisite  number  of  ciphers  to  the  minuend,  or 
conceive  them  to  be  annexed  (Principle  3°,  Art.  88). 

(5.)  (6.)  (7.)                (8.)  (9.) 

13.700  13.7  884.1300  884.13  $8. 

8.299  8.299  33.7865  33.7865  $4^5 

5.401  5.401  8503435  850.3435  $3^5 

10.  From  298.789  subtract  196.493.        Ans.  102.296. 

11.  From  2684.11  subtract  199.8637.       Ans.  2484.2463. 


122  DECIMAL     FRACTI0K8. 

Find  the  difference  between, 

12.  127.334  and  55.827.  21.  ^bft.  and  25.0003//?. 

13.  94.8607  and  27.861.  22.  6  yds.  and  .0006  yds. 

14.  986.444  and  98.6438.  23.  $14,003  and  19.875. 

15.  $17,025  and  $7,255.  24.  13.4072  and  9.1875. 

16.  2.867/if.  and  .9965/i?.  25.  18.65  and  12.0734. 

17.  $661.40  and  $95,472.  26.  17.314  and  12.9921. 

18.  $25,000  and  $1,077.  27.  13.3125  and  8.4139. 

19.  100  yds.  and  99.001  yds,  28.  $34,883  and  $9.43. 

20.  41.02  and  40.021.  29.  87.007/^5. and  10.895/Z>.^. 

30.  $1.87  +  $3.945  +  $27— ($6.42  +  $15.07  +  $.25)=? 

31.  125.6  Ihs.  +  27.42  Z2>s.-  (4.3  Ihs.  + 12.11  Ihs.  +  9  Ihs.)  =? 

32.  $5,000  +  $325,175  —  ($2,710.75 -  $147.56)  =  ? 

33.  ($794.26-$215.875)-($456.375-$211.12)=:? 

PRACTICAL     PROBLEMS, 

1.  Mr.  Holmes  bought  a  cow  for  $45,125  and  sold  her 
for  $49.18;  what  did  he  gain? 

2.  From  a  piece  of  cloth  containing  42.37  yds.,  16.89  yds. 
were  cut  off;  how  many  yards  remained  in  the  piece? 

3.  What  is  the  difference  between  $875,043  and  $704.91  ? 

4.  How  much  must  I  add  to  $617.37^  to  make  $922.75  ? 

5.  A.'s  income  is  $6,250  per  year,  of  which  he  spends 
$3,142.75  and  lays  up  the  rest ;  what  does  he  lay  up  ? 

6.  From  $981.43  +  $456.81  subtract  $498.75. 

7.  From  $10,000  subtract  $4,367.18  +  $3,587.47. 

8.  From  $965  +  $341.60  subtract  $433.33  +  $89.47. 

9.  A  man  received  the  following  sums:  $27.40,  $68.75, 
$810.47,  $386.59,  and  $2.20;  he  paid  out  the  following 
sums:  $78.67,  $129.72,  $119.46,  and  $3.88;  how  much 
had  he  left? 


SUBTRACTION".  133 

10.  A.  had,  at  the  beginning  of  the  year,  goods  worth 
$10,500 ;  during  the  year  he  bought  goods  to  the  amount 
of  $9,345.75,  and  sold  to  the  amount  of  $13,450.95;  at  the 
close  of  the  year  he  had  goods  worth  $11,122.37 ;  how 
much  did  he  make  during  the  year  ? 

11.  A  lady  bought  a  dress  for  $42.18,  a  bonnet  for 
$17.65,  and  a  pair  of  gloves  for  $1.87|- ;  she  gave  for  them 
a  $100  bill ;  how  much  change  ought  she  to  receive  ? 

12.  A.  bought  37.41  cords  of  wood,  of  which  he  sold 
8.3 61  and  burned  13.426 C. ;  how  many  cords  had  he  left? 

13.  A  flagstaff  is  made  up  of  two  parts,  the  upper  part 
being  27. 84= ft.  long,  and  the  lower  part  57.86/^.  long:  now 
if  the  lower  part  is  set  11.31/if.  in  the  ground,  how  many 
feet  of  the  whole  staff  is  above  the  ground  ? 

14.  A  man  had  $137.26,  of  which  he  spent  $17. 87 J  for 
coal,  $22.12i  for  flour,  $7.42  for  soap,  and  $32.79  for  a  suit 
of  clothes  ;  how  much  had  he  left  ? 

15.  From  a  hogshead  of  sugar  containing  397.25  lbs.,  a 
grocer  sold  parcels  as  follows :  110.25  lbs.,  64.5  Z^s.,  14.25  lbs,, 
29.375  lbs.,  39.23  lbs.,  and  16.33  lbs. ;  how  much  was  left  ? 

16.  A.  is  to  travel  bdHi^niles  in  3  days ;  the  first  day  he 
travels  196.4:miles,  and  the  second  day  he  travels  201.25 
miles  :  how  many  miles  must  he  travel  the  third  day  ? 

17.  A  farmer  had  a  colt  worth  $147^,  which  he  traded 
for  a  cow  worth  $42,375,  4  calves  worth  $22|,  and  the 
balance  in  cash  ;  how  much  cash  did  he  receive  ? 

18.  A  merchant  bought  a  piece  of  cloth  for  $75f,  a  box 
of  ribbons  for  $25|,  and  a  quantity  of  thread  for  $27.87  ; 
he  sold  the  cloth  for  $87,125,  the  ribbons  for  $22.16,  and 
the  thread  for  $21f :  did  he  gain  or  lose,  and  how  much  ? 


124  DECIMAL     FRACTION^. 

MULTIPLICATION    OF    DECIMALS. 
DEFINITION. 

95.  Multiplication  of  Decimals  is  the  operation  of 
iinding  the  product  of  two  decimals. 

MENTAL.      EXERCISES. 

1.  If  a  copy-book  costs  2  tenths  of  a  dollar,  what  will  4 
copy-books  cost?  What  is  4  times  2  tenths  of  a  dollar? 
What  is  6  times  2  tenth's?  How  many  units  and  how 
many  tenths  in  the  product?  How  many  tenths  are  6 
times  9  tenths  9  How  many  units  and  how  many  tenths 
in  the  product  ? 

2.  If  a  melon  costs  3  tenths  of  a  dollar,  what  does  1 
tenth  of  a  melon  cost?  What  is  1  tenth  oi  3  tenths? 
What  is  the  product  of  .3  by  .1  ?  of  .3  by  .2  ?  of  .3  by  ,5  ? 
of .  7  by  .  9  ?  of .  8  by  .  8  ?  What  is  the  decimal  unit  of  the 
product  of  tenths  by  tenths  9 

3.  What  is  1  tenth  of  4  hundredths  9  What  is  the  pro- 
duct of  4  hundredths  by  1  ^ewjf^  9  of  .04  by  .2  ?  of  .04  by  .4  ? 
of  .05  by  .7  ?  of  .09  by  .9  ?  What  is  the  decimal  unit  of 
the  product  OiX hundredths  by  tenths9 

4.  What  is  the  product  of  .005  by  .2  ?  of  .007  by  .09  ? 
of  .006  by  .008  ?  of  .7  by  .6  ?  of  .7  by  .04  ?  of  .03  by  .07  ? 

5.  What  is  one  hundredth  of  one  hundredtli  ?  What  is 
the  product  of  3  hundredths  by  7  hundredths  9  What  is 
the  decimal  unit  of  the  product  of  hundredths  by  hun- 
dredths 9    What  is  the  product  of  .09  by  .06  ?  of  .07  by  .08  ? 

Note. — Multiplication  of  decimals  depends  on  the  same  principles 
as  multiplication  of  integers,  and  also  on  the  rule  for  the  multiplica- 
tion of  comm,(m  fractions.,  (Art.  79). 


MULTIPLICATION.  125 

OPERATION      OF     MULTIPLICATION. 

96.  Let  it  be  required  to  find  the  product  of  7.8  and 
.82. 

Explanation. — Here  the  decimals  opebation. 

are  first  changed,  to  equivalent  common  78         82 

fractions  and  multiplied  together  by         '-o  X  .o/C  =  --  x  jx^ 
the  rule  of  Article  7^) ;   the  resulting 

fraction  is  then  reduced  to  the  decimal         __  __  (3,395^ 

form  ;  in  doing  this,  we  have  actually  1000 

multiplied  the  given  decimals  together, 

without  reference  to  their  decimal  points,  and  in  the  result  we  have 
pointed  off  as  many  decimal  figures  as  there  are  in  both  factors. 

Since  all  similar  cases  may  be  treated  in  the  same  manner,  we 
have  the  following 

RULE. 

Multiply  as  in  simple  numbers,  and  point  off, 

frojn  the  right  of  the  product,  as  many  decUnal 

figures  as  there  are  in  both  factors. 

Note. — If  the  number  of  places  in  the  product  is  less  than  the 
number  of  decimal  places  in  both  factors,  prefix  as  many  ciphers  as 
may  be  necessary. 

EXAM  PLES. 

1.  What  is  the  product  of  3.05  by  4.102  ? 

Ans.  12.5111. 

2.  What  is  the  product  of  .003  by  .042  ? 

Ans.  .000126. 
Perform  the  following  multiplications  : 

3.  $38.4  by  16.7.  9.  .0463 /^>5.  by  .0081. 

4.  -$14.25  by  .375.  10.  701.005  x  60.06. 

5.  1,500  ^>?^  by  .00014.  11.  456.05  x  3.825. 

6.  $1,009  by  .0012.  12.  308.25  x  .0775. 

7.  146.05  ?/^s.  by  128.6.         13.  27.032  x  14.3. 

8.  Mhft.  by  .16.  14.  $380.06  x  22. 


126  DECIMAL     FRACTION'S. 

15.  $24.07  X  .125.  25.  0.0156  rods  x  6.75. 

16.  t75  X  .33.  26.  .2897  x  3020. 

17.  $456.87  X  .066.  27.  $37.55  x  45.64. 

18.  $798,007  X  .08.  28.  3.005  x  21.82  x  14.71. 

19.  $.034  X  .08.  29.  8.013  x  11.7  x  0.774. 

20.  1lA6lbs.  X  2.7504.  30.  12.12  x  300.7  x  8.004. 

21.  42.2  X  2.004.  31.  0.713  x  2.346  x  2.005. 

22.  79.004  X  .00473.  32.  $12.5  x  7.2  x  16.5. 

23.  412.5384  x  1.00003.  33.  4:.2  lbs.  x  8.1  x  2.4. 

24.  ^OMyds,  x  .00293.  34.  1.7  yds.  x  11.4  x  82.3. 

Note. — To  multiply  a  decimal  or  a  mixed  decimal  by  10,  100, 
1,000,  etc.,  move  tlie  decimal  point  as  many  places  to  tlie  right  as 
there  are  ciphers  in  the  multiplier,  annexing  ciphers  to  the  multi- 
plicand if  necessary. 

35.  What  is  the  product  of  77.56  by  10?    Ans.  775.6. 

36.  What  is  the  product  of  .0075  by  100  ?      Ans.  .75. 

37.  What  is  the  product  of  6.6  by  1000  ?     Ans.  6600. 

38.  ($31.45  +  $18.2)  x  7.2  —  $240.15  =  ? 

39.  {ISO.Qlbs.  -  SQAlbs.)  x  (67.2  -  3.47)  =  ? 

40.  $150.75  x  16.3  +  $211.5  x  16  —  $114.25  x  9  =  ? 

41.  (dS.4:yds.  +  5Q.4:yds.)  x  7.2  -  18.36^^^/5.  x  8.1  =  ? 

42.  (463.45  +  31.4  -  2.175)  x  (18.2  —  11.07)  =  ?    . 

PRACTICAL     PROBLEMS. 

1.  What  is  the  cost  of  17  barrels  of  flour  at  $6.37^  a 
barrel  ? 

2.  Of  85^  lbs.  of  tea  at  $1.37i  a  pound  ? 

3.  Of  311  yds.  of  linen  at  Q4^cts.  a  yard? 

4.  Of  41^  gallons  of  wine  at  $3.12|  a  gallon  ? 

5.  Of  278  cords  of  wood  at  $9.62|  a  cord  ? 
.  6.  Of  17  lbs.  of  tea  at  $.75  a  pound  ? 


MULTIPLICATION.  127 

7.  Of  7.5  reams  of  paper  at  $3.62|-  a  ream  ? 

8.  Of  2,754  sheep  at  $5,121  apiece  ? 

9.  Of  47.75  bu,  of  corn  at  $.875  a  bushel  ? 

Note. — Let  the  student  apply  the  rule  for  approximate  results 
explained  in  Art.  90,  finding  values  to  the  nearest  mill.  To  secure 
uniformity  the  rule  should  be  applied  at  each  step  of  the  operation. 

10.  A  grocer  sold  25.5  lbs.  of  sugar  at  12^  cts.  a  pound, 
and  18.6  lbs.  of  lard  at  Id^cts.  a  pound ;  how  much  did  he 
receive  for  both  ? 

11.  A  farmer  sold  Sl^bu.  of  oats  at  4:2^  cts.  a  bushel, 
and  35:^  bu.  of  potatoes  at  37-J  cts.  a  bushel ;  he  received 
for  the  same  4:3^ yds.  of  muslin  at  12^  cts.  a  yard,  and  the 
balance  in  cash :  how  much  cash  did  he  receive  ? 

12.  A.  sold  75  bu.  of  wheat  at  I1.12J-  a  bushel,  36.2  bu. 
of  beans  at  $2,374  ^  bushel,  and  97^  lbs.  of  butter  at  22^  cts. 
a  pound ;  what  did  he  receive  for  the  whole  ? 

13.  A  man's  wages  are  $18,874  ^  week,  and  his  expenses 
are  $13.25  a  week ;  how  much  can  he  save  in  14^  weeks  ? 

14.  A  man  was  to  walk  245|  in  7  days :  for  the  first 
3  days  he  walked  at  the  rate  of  34.36  miles  a  day,  and  for 
the  next  three  days  he  walked  36.75  miles  a  day;  how  far 
had  he  to  walk  the  seventh  day  ? 

15.  The  distance  from  St.  Louis  to  New  Orleans  is  1332 
miles ;  tAvo  boats  start  at  the  same  time,  one  from  St.  Louis , 
and  the  other  from  ISTew  Orleans,  are  run  towards  each 
other ;  the  boat  from  St.  Louis  makes  230f  miles  a  day, 
and  the  one  from  New  Orleans  196|  miles  a  day:  how  far 
apart  are  they  at  the  end  of  2^  days  ? 

16.  A  man  starts  from  a  certain  point  and  travels  in  a 
certain  direction  at  the  rate  of  7.25  miles  an  hour ;  at  tlie 


128  DECIMAL     FRACTIOiq"S. 

end  of  2J  hours  a  second  man  starts  from  the  same  point 
and  travels  in  an  opposite  direction  at  the  rate  of  6.29 
miles  an  hour :  how  far  apart  are  they  at  the  end  of  the 
sixth  hour  ? 

17.  A  carpenter  earned  $12.87^  a  week  for  3  weeks;  the 
first  week  he  spent  18.333,  the  second  week  he  spent  $9.18, 
the  third  week  he  spent  $7|,  and  the  rest  he  saved  ;  how 
much  did  he  save  ? 

18.  A  gardener  sold  his  cabbages  for  1212.87^,  and  his 
turnips  for  $118.33 ;  the  cost  of  raising  the  cabbages  was 
1119.75,  and  the  cost  of  raising  the  turnips  was  $99.87^: 
what  was  his  profit  on  the  two  crops  ? 

19.  A  man  bought  43  sheep  at  the  rate  of  4  dollars  and 
67i  cents  a  piece,  and  sold  the  lot  for  215  dollars  and 
42^  cents  ;  did  he  gain  or  lose,  and  how  much  ? 

20.  A  man  made  a  journey  as  follows:  he  traveled 
7j  hours  by  rail  at  the  rate  of  22.75  miles  an  hour, 
9-|  hours  by  stage  at  the  rate  of  6.75  miles  an  hour,  and 
11.75  hours  on  foot  at  the  rate  of  4.62  miles  an  hour; 
what  was  the  length  of  the  journey  ? 

DIVISION     OF     DECIMALLS. 
DEFINITION. 

97.  Division  of  Decimals  is  the  operation  of  find- 
ing the  quotient  of  one  decimal  by  another. 

MENTAL     EXERCISES. 

1.  What  is  the  product  of  .3  by  .5  ?  What  then  is  the 
quotient  of  .15  by  .5  ?  How  many  decimal  places  in  the 
dividend?  in  the  divisor?  in  the  quotient? 

2.  What  is  the  product  of  .12  by  .13  ?    What  then  is 


DIVISION^.  129 

the  quotient  of  .0156  by  .12  ?  by  .13  ?  How  many 
decimal  places  in  the  dividend?  in  the  divisor?  in  the 
quotient  ? 

3.  What  is  the  product  of  .003  by  .9  ?  What  then  is 
the  quotient  of  .0027  by  .9  ?  How  does  the  number  of 
decimal  places  in  the  quotient  compare  with  the  number 
in  the  dividend  and  in  the  divisor. 

Note. — Division  of  decimals  depends  on  the  same  principles  as 
division  of  integers,  and  also  on  the  rule  for  the  division  of  frac- 
tions (Art.  81). 

OPERATION    OF    DIVISION. 

98.   Let  it  be  required  to  divide  7.8  by  .125. 

Explanation. — Here  we  opebation. 

have  reduced  the  given  ded-  •j'^g        'j'g        ]^25 

mals  to    common  fractions 


A  A'   'A  Ay.    ,x.         1       *  -125        10    •    1000 

and  divided  by  the  rule  of 

Art.  81  ;  we  have  then  re-         _^  78       1000  _  78000 

duced  the  result  to  the  deci-         —  10  ^    105   —    105 — '~ 

mal  form ;  in  doing  this,  we 

have  actually  annexed  three  =  624  -7-  10  =:  62.4 

decimal  ciphers  to  the  divi 

dend  (Art.  88),  and  divided  the  result  by  the  divisor,  without  ref- 

erence  to  the  decimal  point ;    then  from  the  quotient   we  have 

pointed  off  as  many  decimal  figures  as  the  number  in  the  reduced 

dividend  exceeds  that  in  the  divisor. 

All  similar  cases  may  be  treated  in  the  same  manner ;  hence, 

the  following 

RULE. 

Annex  decimal  ciphers  to  the  dividend  if  neces- 
sary;  then  divide  as  in  simple  numbers  and^  point 
off  from  the  right  of  the  quotient  as  many  deci7)^al 
figures   as   the  number   of  decimal  places  in  the 

dividend  exceeds  that  in  the  divisor. 
5 


130  DECIMAL     FRACTIONS. 

Notes. — 1.  The  dividend  must  contain  as  many  decimal  figures 
as  the  divisor,  but  it  may  contain  more.  If  the  number  of  decimal 
figures  is  the  same  in  both,  the  quotient  is  a  whole  number. 

2,  If  the  number  of  figures  in  the  quotient  is  less  than  that  re- 
quired by  the  rule,  a  sufficient  number  of  ciphers  must  be  prefixed. 

EXAMPLES. 

1.  Divide  40.05  by  45.  Ans.  8.9. 

2.  Divide  .0141  by  .00047.  Ans,  30. 

3.  Divide  2.3  by  1437.5.  Ans.  .0016. 
Perform  the  following  indicated  divisions,  limiting  ap- 
proximate values  to  the  fourth  decimal  place : 

4.  .00125  -^  .5.  15.  15.875  -f-  35.25. 

5.  $34.75  -^  25.  16.  480  —  3.12. 

6.  46.103  -j-  2.14.  17.  $1.8  -^  28.8. 

7.  7.8125  ^  31.25.  18.  17.1031  yds,  ^  .63. 

8.  $2756.25  -^  31.5.  19.  .09925  -^  .37. 

9.  $68,875  -f- 14.5.  20.  3.72812  ^  4.07. 

10.  3414.52  -^  30.25.  21.  $18.1771  -^  27.13. 

11.  16.025  -r-  .045.  22.  101.6688  -~  43.08. 

12.  .9375 /«J.  -T-  .075.  23.  1.51088  -^  .019. 

13.  112.1184  -^  9.16.  24.  187.12264  ^  1.52. 

14.  9322.15  -^  6.275.  25.  $71.1022  -^  $9.43. 

Note.— To  divide  a  decimal  or  a  mixed  decimal  by  10,  100,  1000, 
&c.,  move  the  decimal  point  as  many  places  to  the  left  as  there  are 
ciphers  in  the  divisor,  prefixing  ciphers  to  the  dividend  if  necessary. 

26.  What  is  the  quotient  of  77.56  by  10  ?    Ans.  7.756. 

27.  What  is  the  quotient  of  .0075  by  100  ?  Ans.  .000075. 

28.  What  is  the  quotient  of  6.6  by  1000?    Ans.  .0066. 

29.  ($28  +  $11.75)  -r- 1.25  -f  $38.75  =  ? 

30.  $50  -^  5.75  -f  ($10  -  $3.75)  x  1.2  =  ? 

31.  ($13.75  -  $1.87i)  -^  (12.75  -  4.5)  =  ? 


DIVISION.  131 


(63.5/if.  -  24.25/0  -r-  (17.25  -  11.75)  =  ? 
uu.  4^7/ds.  X  2.2  +  lliyds.  -^  1.25  +  IS.Sl 6  yds.  = 
34.  (47.3  Z^5.  +  6.7  lbs.)  -^  (34.18  -  16.78)  =  ? 


32 

33.  4^2/^ 


PRACTICAL     PROBLEMS. 

1.  If  a  man  can  earn  $519.75  in  13.5  weeks,  how  much 
can  he  earn  in  1  week  ?  A?is.  $38.50. 

2.  If  20.5  bic.  of  buckwheat  cost  $12.71,  what  is  the  cost 
of  1  bu.  ?  of  7i  bii.  ?  Atis.  62  cts. ;  $4.65. 

3.  If  a  barrel  of  flour  costs  15.75,  how  many  barrels 
can  be  bought  for  $1,035  ?  A7is.  180  bbls, 

4.  If  1,000  acres  of  land  cost  $17,586,  what  is  the  cost 
of  1  ^.  ?  of  75i  A.  ?  Ans.  $17.58^^ ;  $1,32.743. 

5.  If  1  acre  costs  $25.62,  how  many  acres  can  be  bought 
for  $1,242.57? 

6.  K  75.3  cords  of  wood  cost  $640.05,  how  much  will 
1(7.  cost?  6.3  a? 

7.  If  1  cord  of  wood  costs  $8.25,  how  much  wood  can 
be  bought  for  $156. 75  ?     For  $30.52^  ? 

8.  At  $4.28  a  yard,  how  much  cloth  can  be  bought  for 
$74.90  ?    How  much  for  $152.52^  ? 

9.  A  farmer  sold  27.5  lbs.  of  butter  at  20  cts.  a  pound,  and 
17.5  bu.  of  oats  at  75  cts.  a  bushel ;  he  took  in  payment  in 
sugar  at  12^ cts.  a  pound:  how  many  pounds  did  he 
receive  ? 

10.  There  are  31.5  gallons  in  a  barrel;  how  many  bar- 
rels are  there  in  2756.25  gallons  ? 

11.  There  are  l,"^ 60 yds.  in  1  mile;  how  many  miles  are 
there  in  23,760 yds.?    In  26,840  yds.? 

12.  A  merchant  buys  a  piece  of  cloth  containing  35  yds. 


132  DECIMAL     FRACTIONS. 

for  $87.50;   he  wishes  to  sell  it  so  as  to  gain  117.50:  at 
what  price  must  he  sell  it  per  yard  ? 

13.  A  man  can  travel  4:4:1.5 miles  in  7.5  days;  how  far 
can  he  travel  in  1  day  ?  in  9  J  days  ? 

14.  A  man  travels  7.25  days  at  the  rate  of  211.5  miles  a 
day;  on  his  return  he  makes  the  whole  journey  in  5  days: 
how  many  miles  is  that  per  day  ? 

15.  A.  and  B.  start  at  the  same  time  from  points  147.16 
miles  apart,  and  travel  toward  each  other  till  they  meet ; 
if  A.  travels  at  the  rate  of  7.75  mi.  an  hour,  and  B.  at  6.4  mi. 
an  hour,  how  long  before  they  meet  ? 

16.  A  speculator  bought  78.25  acres  of  land  for  19,781.25, 
and  sold  it  so  as  to  gain  13.50  an  acre ;  what  did  he  get 
per  acre  ? 

17.  If  621!  bu.  of  wheat  cost  1592.87^,  what  is  the  cost 
o^lbu.f  of  6.6  bu.  9 

18.  If  115  lbs.  of  beef  cost  $19.89^,  what  will  93  lbs.  cost 
at  the  same  rate  ? 

19.  If  I  pay  $39.48  for  28  bu.  of  wheat,  what  must  I 
pay  for  48  bushels  at  the  same  rate? 

20.  A  grocer  bought  114  gallons  of  vinegar  at  22^  cts.  a 
gallon,  and  sold  it  so  as  to  gain  $7.98  ;  at  what  rate  per 
gallon  did  he  sell  it  ? 

21.  A.  starts  from  a  certain  place  and  travels  along  a 
road  at  the  rate  of  4.66  miles  an  hour ;  B.  starts  13.75 
miles  behind  him  and  travels  in  the  same  direction  at  the 
rate  of  5.91  miles  an  hour:  how  long  before  B.  will  over- 
take A.  ? 

22.  A  farmer  sold  22.5  bu.  of  wheat  at  $1.18  a  bushel, 
and  a  certain  number  of  bushels  of  oats  at  68  cts.  a  bushel ; 


DIVISION.  133 

he  received  for  his  oats  $22. 41  more  than  he  did  for  his 
wheat :  how  many  bushels  of  oats  did  he  seU  ? 

23.  If  mibs.  of  coffee  cost  $2.07,  what  will  lib.  cost? 
What  will  31 J /^5.  cost? 

24.  How  many  bushels  of  oats  at  62^  cts.  a  bushel  will 
pay  for  4J  thousand  of  lumber  at  17.50  a  thousand  ? 

25.  A  farmer  exchanged  70  Ui.  of  rye  at  $0.92  a  bushel, 
for  40  hi.  of  wheat  at  11.371-  a  bushel,  and  the  balance  in 
oats  at  10.40  a  bushel ;  how  many  bushels  of  oats  did  he 
receive  ? 

26.  If  a  man  can  travel  32.48  miles  in  .8  of  a  day,  how 
far  can  he  travel  in  5.3  days  ? 

27.  What  is  the  sum  of  the  quotients  of  24  by  9.6,  of 
42.75  by  11.4,  and  of  17.85  by  4.2  ? 

28.  A.  and  B.  start  together  and  travel  in  the  same 
direction  around  an  island  whose  circuit  is  4.2  miles  ;  A. 
travels  at  the  rate  of  4.6  mi.  an  hour,  and  B.  at  the  rate 
of  5.2 mi,  an  hour:  how  many  hours  before  they  are 
together  again  ? 

29.  A  man  bought  a  farm  containing  64.5  acres  for 
$1,773.75  ;  what  was  that  per  acre  ? 

30.  How  many  cords  of  wood  at  $8  a  cord  must  be  paid 
for  24  yards  of  cloth  at  $3.50  a  yard  ? 

31.  If  60  bushels  of  turnips  cost  $18.60,  how  much  will 
19  bushels  cost  ? 

32.  If  10  tons  of  coal  cost  $57.50,  how  many  tons  can 
be  bought  for  $235.75  ? 

33.  A  tailor  cuts  from  31.25  yds.  of  cloth  6  coats,  each 
taking  3.75  pds.,  and  makes  the  rest  into  vests,  each  taking 
1.25  yds.  ;  how  many  vests  does  he  make  ? 


134 


BUSINESS     OPERATIONS. 


III.   CONTRACTIONS  AND  BUSINESS 
OPERATIONS. 

ALIQUOT     PARTS. 

99.  An  Aliquot  Part  of  a  number  is  one  of  the  equal 
parts,  whether  integral  or  fractional,  into  which  the  num- 
ber can  be  divided. 

The  principal  aliquot  parts  of  a  dollar  are  shown  in  the 
following 

TABLE. 

50    cts.,  equal  to  J  of  II.         12|^  ds.,  equal  to  \  of  $1. 

33i  ds.,      ''  ''  i  ''   '' 

25    ds.,     ''  "  \  ''   '' 

20    ds.,     ''  ''  \  ''   '' 


10    ds.. 

a 

"tV"  " 

6  J  ds.. 

a 

"t^"   " 

5    ds,, 

(( 

"^•' " 

100.  To  find  the  cost  of  any  number  of  things  when 
1  thing  costs  an  aliquot  part  of  a  dollar,  we  have  the 
following 

RULE. 

Divide  the  number  of  things  by  the  number  of 
times  the  price  of  one  thing  is  contained  in  $1; 
the  quotient  will  be  the  required  number  of  dollars. 

EXAM  PLES. 

1.  What  is  the  cost  of  64  bushels  of  oats  at  50  cents  a 
bushel?  Ans.  1-^  =  $32. 

2.  What  will  116  pounds  of  beef  cost  at  20  cents  a 
pound  ?  Ans.  $J^  =  -^23.20. 

3.  Of  250  melons  at  25  ds.  each  ? 

4.  Of  144  pencils  at  VZ^cts.  each  ? 

5.  Of  75  oranges  at  5  ds.  each  ? 


ALIQUOT   PARTS.  135 

6.  Of  69  yds.  of  sheeting  at  33i  ds.  a  yard  ? 

7.  Of  50  dozen  marbles  at  6J  cts.  a  dozen  ? 

8.  Of  73  lbs.  sugar  at  12J  cts.  a  pound  ? 

9.  17  ^o;2e;i  eggs  at  25  cts.  a  dozen  ? 

10.  Of  117  ^w.«r^5  berries  at  20  cts.  a  quart  ? 

11.  Of  47  lbs.  coffee  at  ^?>^cts.  a  pound  ? ' 

12.  Of  145  lbs.  rice  at  6J  cts.  a  pound  ? 

13.  Of  87.3  Ihs.  coffee  at  33^  cts.  a  pound  ? 

14.  Of  315  lbs.  sugar  at  12^  cts.  a  pound  ? 

*    15.  Of  70  lbs,  of  butter  at  33^  cts.  a  pound  ? 
16.  Of  35  doz,  eggs  at  20  cts.  a  dozen  ? 

101.  To  find  the  cost  of  things  sold  by  the  hundred, 
or  by  the  thousand,  we  have  the  following 

RULE. 

Multiply  the  cost  of  100,  or  1000  things,  hy  the 
mnnher  of  things,  and  move  the  decimal  point  2, 
or  3  places  to  the  left. 

EXAMPLES. 

1.  What  is  the  cost  of  460  oranges  at  $3.50  joer  hundred  ? 

.        $3.50x460      ._,_ 
Ans.  j^r =  $16.10. 

2.  Of  1,726/j?.  of  boards  at  $3  per  1,000 /if.  ? 

3.  Of  47,555  bricks  at  $7.50  per  1,000  ? 

4.  Of  freight  on  8,714  lbs.  at  62^  cts,  a  hundred? 

5.  Of  83,750/i^.  of  stone  at  $60  a  thousand  9 

6.  Of  763  lbs,  of  pork  at  $4.50  a  hundred  9 

7.  Of  511  lbs.  of  beef  at  $7  a  hundred  9 

8.  Of  1,432  lbs.  of  pork  at  $8.25  per  hundred  9 


136  BUSINESS     OPERATIONS. 

9.  Of  8,741 /if.  of  plank  at  $30  per  thousand  f 

10.  Of  4,875//^.  of  boards  at  $17  per  thousand  f 

11.  Of  7,320  papers  tacks  at  $30  a  thousand  9 

12.  Of  756/if.  of  stone  at  $5  a  hundred? 

13.  Of  3,450  oysters  at  $2  a  hundred? 

14.  Of  7,846  lis.  of  hay  at  dOcts,  a  hundred? 

102.  To  find  the  cost  of  things  sold  by  the  ton,  that 
is,  by  the  two  thousand  pounds,  we  have  the  following 

RULE. 

Multiply  half  the  cost  of  a  ton  hy  the  number  of 
pounds,  and  move  the  decimal  point  in  the  product 
three  places  to  the  left. 

EXAMPLES. 

I.  What  is  the  cost  of  3,475  lis.  of  plaster  at  $7.50  per 

.      o                                   A        ^3.75x3475       .^_„^ 
ton  ?  Ans.   j^^^ =  $13.03|. 

2.  Of  transporting  6,742  lbs.  at  $7  a  ton  ? 

3.  Of  6,527  Ihs.  of  oats  at  $30J  a  ton  ? 

4.  Of  18,747  lbs.  of  coal  at  $8|  per  ton  ? 

5.  Of  8,142  lbs.  of  iron  at  $100  a  ton  ? 

6.  Of  3,120  lbs.  of  wool  at  $660  a  ton  ? 

7.  Of  1,620  lbs.  of  hay  at  $16  per  ton  ? 

8.  Of  5,782  lbs.  of  pig  iron  at  $22  a  ton  ? 

9.  Of  7,711  lbs.  of  straw  at  $15  per  ton  ? 
10.  Of  8,824  lbs.  of  hay  at  $15  per  ton  ? 

II.  Of  3,509  lbs.  of  wheat  at  $40  a  ton  ? 

12.  Of  2,250  lbs.  of  coal  at  $7.40  a  ton  ? 

13.  Of  14,710  lbs.  of  crushed  stone  at  $3  a  ton? 

14.  Of  16,318  lbs.  of  meal  at  $37  a  ton  ? 


BILLS     AND     ACCOUNTS. 


137 


BILLS     AND     ACCOU  NTS. 

103.  A  bill  is  a  written  statement  of  goods  sold,  ser- 
vices rendered,  or  money  paid,  with  the  date  of  each  item. 

The  ordinary  form  of  a  bill  of  items  is  shown  below,  in 
which  @  stands  for  at. 


i^/c/e^  S^  '^o. 


y/  ^■a'C^.4'i€id.  ix^ ^-e-ci    .     .     .     .     @  ^0.  fS 

^^fS 

So 

7^ 

/4 

^■^&  ^■044.4'l^d    -O^  dU'O.lZ^       .       .       .      @           .-/^ 

^s 

Ss 

O^^^^^fiS^f^-^ 

/(^y 

// 

'C^pt/C, 


d    cJ^ciJc/e<n   ^  ^o 


Note.— The  party  that  (me%  money  is  called  a  debtor  (Dr.),  and 
the  party  to  whom  money  is  due  is  called  a  creditor  (Cr.).  When  a 
bill  is  paid,  it  is  receipted  by  writing  the  name  of  the  creditor,  or 
his  authorized  agent,  at  the  bottom  of  the  bill  after  the  words 
received  payment. 

To  find  the  amount,  or  footing  of  a  bill: 
Fi?id  the  amount  of  each  item  separately,  and 
then  find  the  sum  of  the  results. 


138 


BUSINESS     OPERATIOIS^S. 


EXAM  PLES. 

Find  the  footings  of  the  following  bills : 

(1.)  St.  Louis,  Mo.,  July  15,  1877. 


P.    Gr.    BiSSEL, 


Bought  of  L.  Smith. 


2462  feet  of  hemlock  boards  .     . 
5410      "            "            "      .     .    . 
600       "      scantling 

@         $7  per  1000. 
@        $10  per  1000. 
@  $11.75  per  1000. 
@    $1.25  per  100. 
@  15  cts.  per  foot. 

Amt., 
Received  payment. 

Columbus,  0.,  Ai 
Bought  of  A. 

1012      "      plank 

77      "      hewn  timber.    .    .    . 

(3.) 
J.  D.  Smith, 

1 

ig.  4,  1877. 
HiNE. 

32  yds.  silk 

18  yds.  alpaca 

16  yds.  chintz 

42  yds.  muslin 

15  pieces  tape 

3  pairs  gloves 

12  pairs  stockings      .... 

.     @  $2.12i 
@  87i  cts. 
@  24  cts. 
@  21  cts. 
@  14  cts. 
@  $1.94 
@  62i  cts. 

Amt, 
Rec'd  payment. 

S.  L.  Morse, 


(8.)  Cairo,  lU.,  Sept.  5, 1877. 

To  J.  Bristow,  Dr, 


To  labor,  self,  4  days 
man,  6  days 
materials     .     .     . 


((       << 


$4. 
$3. 


$17 


Amount, 
Received  payment. 


BILLS     AND     ACCOUNTS. 


139 


Write  the  following  in  proper  form  and  find  the  foot- 
ings: 

4.  John  Duffie  bought  of  D.  Plant,  April  7, 1877, 17  yds, 
of  calico  @  12|-c/5.,  12  yds,  of  muslin  @  17  cts.,  2^  yds.  of 
linen  @  73  cts.,  and  9  spools  of  thread  at  7  ds.;  what  was 
the  amount  of  the  bill  ? 

5.  Mrs.  Churchill  bought  of  Knapp  &  Co.,  July  29, 1877, 
the  following  items:  1  shawl  @  125.50,  22  yds.  silk  @  $2.25, 
12  yds.  lace  @  82  ds.^  3  ^rs.  gloves  @  11.25,  and  4  pieces 
tape  @  32  c^5. :  what  was  the  amount  of  the  bill  ? 

6.  Mr.  A.  W.  Stoughton  bought  of  Sweet  &  Co.,  July  30, 
1877,  the  following  items:  12  Arithmetics  @  60  ds., 
20  Geogi'aphies  @  72  ds.,  37  Grammars  @  43  ds.,  4  reams 
paper  @  $2.25,  and  3  boxes  pens  @  11.10 ;  what  was  the 
amount  of  the  bill  ? 

BALANCING     ACCOUNTS. 

104.  An  Account  is  a  written  statement  of  items  of 
debt  and  credit.    A  book  in  which  these  items  are  recorded 
in  separate  columns,  is  called  a  ledger. 
The  ordinary  form  of  a  ledger  account  is  shown  below : 
I)r.  John  Irving  in  acct.  with  Henry  Holt.  Cr. 


1877 

$    c. 

1897    i 

$ 

c. 

Apr.  4 

To  Uyds.  silk       @  $1.75 

24  50 

May    1  By  25&M.  corn      @  $1.10 

27 

50 

May  7 

"    16yd8.c\oih     '•     3.50 

56  00 

May  19   "  46/6*.  butter    "      .25 

11 

50 

July  1 

"   42yds.  muslin  "       .16 

6|72 

July  25  Bal.  'due 

49 

34 

July  7 

"     8  pieces  tape  "       .14 

lll2 

"         ' ________ 

Amount 

88i»4 

88 

34 

Note. — The  items  for  which  Henry  Holt  is  indebted  to  John 
Irving,  who  makes  the  account,  are  placed  in  the  column  headed 
Dr.,  and  the  items  for  which  John  Irving  is  indebted  to  Henry  Holt 
are  placed  in  the  column  headed  Or. ;  the  former  are  called  debits, 
and  the  latter  credits.  The  balance  is  the  amount  which,  entered  in 
the  proper  column,  will  make  the  sums  of  the  two  columns  equal. 


140  BUSINESS     OPERATIONS. 

To  find  the  balance  of  a  ledger  account :  Find  the  sum 
of  the  debits  and  of  the  credits  separately,  and 
take  their  difference. 

EXAMPLES. 

Find  the  balance  in  the  following  accounts : 

1.  Debits,  $4.19,  5.25,  7.44,  17.11,  47.42,  110.76,  308.12, 
114.04,  3.79,  and  10.94;  Credits,  $3.43,  2.11,5.27,15.60, 
108.29,  84.12,  216.58,  94.26,  and  13.12. 

2.  Debits,  112.56,  14.92,  27.14,  110.94,  94.11,  83.88,  7.46, 
4.39,  71.18,  14.43,  and  1.94;  Credits,  $14.87,  110.72,  3.69, 
11.18,  7.42, 19.78,  9.94,  12.12,  and  5.55. 

REVIE^AT  QUESTIONS. 
(83.)  What  is  a  decimal  fraction?  (84.)  What  is  a  decimal  ? 
Decimal  point?  (85.)  Repeat  the  numeration  table  for  decimals. 
How  are  decimal  places  counted  ?  Rule  for  writing  a  decimal  ? 
(86.)  Rule  for  reading  a  decimal  ?  (87.)  What  are  the  units  of 
U.  S.  currency?  Their  relation?  How  is  U.  S.  money  written? 
How  read  ?  (88.)  State  the  fundamental  principles  used  in  treat- 
ing fractions.  (89.)  Rule  for  reducing  common  fractions  to  deci- 
mals. (90.)  What  is  the  practical  rule  for  finding  approximate 
values  of  a  fraction  ?  (92.)  Rule  for  addition  of  decimals  ?  (94.) 
Rule  for  subtraction  of  decimals  ?  (96.)  Rule  for  multiplication  of 
decimals?  (98.)  Rule  for  division  of  decimals?  (99.)  What  is 
an  aliquot  part  of  a  number  ?  Name  some  of  the  aliquot  parts  of  a 
dollar.  (lOO.)  How  do  you  multiply  by  an  aliquot  part  of  100  ? 
(lOl.)  How  do  you  find  the  cost  of  things  sold  by  the  hundred  or 
thousand  ?  (102.)  Of  things  sold  by  the  ton  ?  (103.)  What  is  a 
bill?  How  find  its  looting?  (104.)  What  is  an  account?  A 
debtor?  A  creditor?  Debits?  Credits?  How  do  you  find  the 
balance  of  a  ledger  account? 


f>l<! 


POU^B 


>f< 


I,    DEFINITIONS    AND 
TABLES. 

DEFINITIONS. 

105.  A  Denominate  Number  is  one 

whose  unit  is  named;  as,  ^  feet,  b pounds, 

16  pennyweights  (Art.  5). 

Numbers  that  have  the  same  unit  are  of  the  same  denomination  ; 
those  that  have  different  units  are  of  different  denominations.  Thus, 
^feet,  and  1  feet,  are  of  the  same  denomination  ;  dfeet,  and  7  yards, 
are  of  different  denominations. 

106.  A  Compound  Number  is  a  denominate  num- 
ber whose  units  are  of  .the  sams  hind  but  of  different 
denominations;  as,  ^pounds  6  ounces. 

Denominate  numbers  are  of  the  same  Mud,  when  they  can  be 
expressed  in  terms  of  a  common  unit.  Thus,  ^pounds,  and  6  ounces, 
are  of  the  same  kind  because  both  can  be  expressed  in  ounces. 

If  two  denominate  numbers  are  of  the  same  kind,  that  which  has 
the  greater  unit  is  said  to  be  of  the  higher  denomination.  Thus, 
3  pounds  is  of  a  higher  denomination  than  6  ounces. 

SCALES     OF     COMPOUND      NUMBERS. 

107.  The  Scale  of  a  compound  number  is  a  succes- 
sion of  numbers  showing  how  many  times  the  unit  of 
each  denomination  is  contained  in  the  unit  of  the  next 


142  COMPOUND     NUMBERS. 

higher  denomination.  Thus,  in  English  cnrrency,  i  far- 
things make  1  penny,  12  pe7ice  make  1  shilUng,  and  20 
shillings  make  1  pound;  hence,  the  scale  of  Enghsh  cur- 
rency is 

4,        12,        20. 

In  this  scale,  4  connects  farthings  and  pence,  12  connects  pence  and 
shillings,  and  20  connects  shillings  and  pounds. 

In  United  States  currency  the  scale  is 
10,        10,        10,        10. 

The  scale  of  English  currency  is  varying ;  that  of  the  United 
States  is  uniform.  The  scale  of  United  States  currency  is  called  the 
scale  of  tens,  or  the  decimal  scale. 

The  scales  of  the  most  important  compound  numbers 
are  indicated  in  the  following  tables 

TABLES     OF     CUERENCY. 

1°.   UNITED  STATES  CURRENCY. 

108.  The  United  States  currency  was  established  by 
act  of  Congress  in  1792  ;  its  primary  unit  is  1  dollar, 

TABLE. 

10  mills  (m.)  make  1  cent ct 

10  cents  **      1  dime d, 

10  dimes  ''      1  dollar $, 

10  dollars  "      1  eagle B, 

In  business  transactions  the  terms  eagle  and  dime  are  nearly 
obsolete  ;  the  term  mill  is  seldom  used  except  in  official  reports  and 
in  laying  taxes. 

Note. — The  currency  of  the  Dominion  of  Canada  is  decimal,  and 
like  that  of  the  United  States,  it  is  reckoned  in  dollars  and  cents. 


TABLES.  143 

2\      ENGLISHCURRENCY. 

109.  This  is  the  national  currency  of  Great  Britain. 

TABLE. 

4  farthings  {far.  or  qr.)  make  1  penny d, 

12  pence  '^      1  shilling s. 

20  shillings  "      1  pound £. 

21  shillings  "      1  guinea G» 

The  primary  unit  of  this  currency  is  1  pound  sterling,  and  the 
corresponding  coin  is  called  a  sovereign.  The  value  of  the  sovereign 
is  $4.8665. 

Note. — The  sign  £,  like  the  sign  $,  is  written  before  the  number 
to  which  it  refers  ;  thus,  £25. 

3°.      FRENCH    CURRENCY. 

110.  This  is  the  national  currency  of  France. 

TABLE. 

10  centimes  {cent.)  make  1  decime d. 

10  decimes  "      1  franc fr. 

The  primary  unit  of  this  currency  is  1  franc;  its  value  is 
1%.^  cents,  that  is,  $1  is  equal  to  about  5.18  francs.  French  money 
is  usually  expressed  in  francs  and  decimals  of  a  franc. 

TABLES    OF    WEIGHT. 

4°.      TROY    WEIGHT. 

111.  This  is  used  in  weighing  gold,  silver,  and  some 
kinds  of  precious  stones. 

TABLE. 

24  grains  {gr.)  make  1  pennyweight. .  .dwt. 

20  pennyweights  "     1  ounce oz. 

12  ounces  "    1  pound lb.  Tr, 


144  COMPOUND     J^UMBERS. 

Explanation. — The  pound  Troy  is  the  primary  unit  of  weight 
in  Great  Britain  and  the  United  States.  It  was  declared  by  an  act 
of  Parliament,  which  took  effect  in  1826,  that  the  brass  weight  of  one 
pound  Troy,  then  in  the  custody  of  the  Clerk  of  the  House  of 
Commons  should  be  the  standard,  and  that  all  other  weights  should 
be  derived  from  it.  It  was  enacted  that  the  pound  Troy  should  con- 
tain 5, 7Q0  grains,  and  that  7,000  such  grains  should  make  a  pound 
avoirdupois. 

Note. — The  English  system  of  weights  and  measures  has  been 
adopted  by  act  of  Congress,  and  is  the  legal  system  of  the  United 
States. 

2\      APOTHECARIES'    WEIGHT. 

112.  This  differs  from  Troy  weight  in  the  mode  of 
subdividing  the  ounce ;  it  is  used  in  weighing  medicines. 

TABLE. 

20  grains  (gr.)  make  1  scruple ^ 

3  scruples  "     1  dram 3 

8  drams  "     1  ounce | 

12  ounces  "      1  pound S) 

3'.      AVOIRDUPOIS    WEIGHT. 

113.  This  weight  is  used  in  weighing  the  ordinary 
articles  of  trade  and  commerce. 

TABLE. 

16  ounces  (oz.)      make  1  pound lb. 

25  pounds  "     1  quarter  ....    . .    . .  qr, 

4  quarters                "     1  hundredweight  . . .  cwt 
20  hundredweight    "     1  ton T, 

The  primary  unit  is  1  pound,  equal  to  7,000  grains  Troy. 

In  weighing  coarse  articles  liable  to  wastage,  as  coal  at  the  mines, 
and  the  like,  it  is  customary  to  call  112  Ws.  a  hundredweight,  and 
38  £b8.  a  qumrter. 


TABLES.  146 

TABLES    OF    TIME. 

DEFINITION    AND    EXPL  AN  A  T  I  0  N  . 

114.  Time  is  a  measured  portion  of  duration.  Jts 
primary  unit  is  1  mean,  or  average,  solar  day. 

Explanation. — An  astronomical  year  is  the  time  required  for 
the  earth  to  revolve  about  the  sun  ;  but  this  period  does  not  contain 
an  exact  number  of  days ;  hence,  for  civil  purposes,  an  artificial  year 
is  adopted.  The  length  of  the  civil  year  is  sometimes  365  days,  and 
sometimes  366  days,  and  these  are  so  distributed  that  after  a  long 
period  the  a'derage  length  of  the  civil  year  is  equal  to  that  of  the 
astronomical  year. 

Every  year  divisible  by  4  (except  centennial  years  not  divisible 
by  400)  are  leap  years  and  contain  366  days  each ;  all  other  years  are 
common  yea/rs  and  contain  365  days  each. 

TABLE. 

60  seconds  (sec.)  make  1  minute min, 

1  hour hr. 

1  day da, 

1  week wh. 

1  common  year  —  yr. 

1  leap  year yr. 

1  century G. 

The  year  is  divided  into  12  parts  called  months.  Their 
order  and  the  number  of  days  in  each  is  shown  in  the 

TABLE. 

1°.  January 31  days.     7°.  July 31  days. 


60  minutes 

a 

24  hours 

a 

7  days 

tt 

365  days 

<e 

366  days 

<( 

100  years 

<( 

2°.  February. . .  28  or  29 

3°.  March 31 

4°.  April 30 

5°.  May 31 

6°.  June 30 


8°.  August .31  " 

9°.  September 30  " 

10°.  October 31  " 

11°.  November 30  « 

12°.  December 31  « 


In  common  years  February  has  28  days  ;  in  leap  years  39. 


146  COMPOUND     NUMBERS. 

MEASUKES    OF    LENGTH. 

'      DEFINITIONS. 

115.  A  Magnitude  is  anything  that  can  be  measured; 
that  is,  anything  that  can  be  expressed  in  terms  of  a  thing 
of  the  same  kind  taken  as  a  unit. 

116,  A  Line  is  a  magnitude  that  has  length,  without 

breadth  or  thickness. 

A  curved  line  is  one  whose  direction  y>-^  v. 

changes  at  every  point ;  as  GH.  /^  N^ 

A  straight   line  is  one  whose  direc-  cubved  line. 

tion  does  not  change  at  any  point ;   as 

AB.  A B 

Straight  lines  are  parallel  when  they  straight  ldtb. 

have  the  same  direction  ;  as  CD  and  EF.  ^ 

The  length  of  a  line  is  the  number  of  ^ P 

times  it  contains  a  given  straight  line  parallel  lines. 

taken  as  a  unit. 

1°.   LONG  MEASURE. 

11'7.  This  is  used  for  measuring  distances  and  dimen- 
sions of  objects. 

TABLB. 

12    inches  {in.)  make  1  foot ft. 

3    feet  "      1  yard yd. 

5 J  yards  **      1  rod rd. 

40    rods  *'      1  furlong . .  .fur. 

8    furlongs,  or  320  rods  "      1  mile mi. 

3    miles  "      1  league lea. 

It  is  found  convenient  in  practice  to  reduce  yards,  feet,  and  inches 
to  Jialf  yards  and  inches  ;  we  thus  avoid  the  inconvenience  of  a  scale 
in  which  one  of  the  numbers  is  fractional.  In  this  case,  the  first 
part  of  the  preceding  table  may  be  replaced  by  the  following : 

18  inches       make  1  half  yard hf.  yd. 

11  half  yards    "      1  rod rd. 


TABLES.  147 

The  yard  is  the  primary  unit  of  English  and  American  measures 
of  length.  By  the  act  of  Parliament  already  referred  to,  it  was 
declared  that  the  brass  standard  yard,  then  in  the  custody  of  the 
Clerk  of  the  House  of  Commons,  should  be  the  imperial  standard 
yard.     From  it  we  derive  all  other  measures  of  length. 

Note. — In  measuring  doth,  ribbons,  and  the  like,  the  yard  is  sub- 
divided into  halves,  quarters,  eighths,  and  sixteenths. 

2°.      SURVEYORS'      MEASURE. 

118.  This  is  used  in  measuring  land.  The  unit  is  a 
Gunter's  Chain ;  this  chain  is  4  rods,  or  66  feet,  in  length, 
and  is  divided  into  100  equal  parts,  called  links. 

TABLE. 

7.92  inches  make  1  link U, 

100       links        "      1  chain ch, 

80       chains     "      1  mile . .  ..mi. 

MEASUKES    OF    SUKFAOE. 

DEFINITIONS. 

119.  A  Surface  is  a  magnitude  that  has  length  and 
breadth,  without  thickness. 

A  Plane  is  a  surface  such  that  if  a  straight  line  is  applied  to  it  in 
any  direction  it  will  coincide  with  the  surface  throughout. 

130.  An  Angle  is  the  opening  between  two  lines  that 
meet  at  a  point ;  as,  BAG. 

The  lines  AB  and  AC  are  called  sides,  and 
the  point  A  is  called  the  vertex  of  the  angle. 

If  one  straight  line  meets  another  so  as  to 
make  the  two  adjacent  angles  equal,  the  first 
line  is  said  to  be  perpendicular  to  the  second, 
and  the  angles  are  called  right  angles  ;  thus, 
if  the  angles  Bx\D  and  BAC  are  equal,  they 
are  both  right  angles  and  BA  is  perpendicular 


D  A  C 

*^  ^^-  BI6HT  ANaiiES. 


148  COMPOtJN'D     NUMBERS. 

121.  A  Square  is  a  plane  figure  bounded  by  four  equal 

lines,  called  Sides,  and  having  all  its 

angles  right  angles ;  as,  ABCD. 

A  Rectangle  is  bounded  by  four  lines,  paral- 
lel two  and  two,  and  having  all  its  angles  right 


angles.  square. 

132.  A  Unit  of  Surface  is  a  square  whose  sides  are 
equal  to  the  unit  of  length.  If  each  side  is  a  yard,  the 
square  is  called  a  square  yard. 

The  Area  of  a  Surface  is  an  expression  for  that  sur- 
face in  terms  of  a  square  unit. 

Note. — It  is  shown  in  geometry  that  the  area  of  a  rectangle  is 
equal  to  the  product  of  its  length  by  its  breadth  ;  that  is,  the  number 
of  square  units  in  the  surface  is  equal  to  the  number  of  units  in  its 
length  multiplied  by  the  number  of  units  in  its  breadth. 

1°.  SQUARE  MEASURE. 

123.  This  is  used  in  measuring  surfaces. 

TABLE. 

144    square  inches  {sq.  in.)  make  1  square  foot sq.  ft. 

9    square  feet  "     1  square  yard . .  ..sq.  yd. 

30J-  square  yards  "      1  square  rod sq.  rd. 

160    square  rods  "     1  acre A. 

It  is  found  convenient  in  practice  to  reduce  square 
yards,  square  feet,  and  square  inches,  to  quarter  square 
yards  and  square  inches,  to  avoid  using  a  scale  containing 
a  fractional  number.     In  this  case 

324  square  inches    make  1  quarter  square  yard . .  qr.  sq.  yd, 

121  square  quarters    ''     1  square  rod sq.  rd. 

160  square  rods  "     1  acre A, 


TABLES. 


149 


2°.      LAND      MEASURE. 

124.  This  is  used  in  measuring  land. 

TABLE. 

10000  square  links  (sq.  U.)  make  1  square  chain sq.  ch. 

10  square  chains  *^     1  acre A. 

640  acres  "     1  square  mile sq.mL 

The  area  of  land  is  also  reckoned  by  the  following 

TABLE. 

40  square  rods  (sg.  rds.)  make  1  rood B. 

4  roods  *^      1  acre A. 

In  governmeut  surveys,  a  square  mile  is  called  a  section  ;  36  sec- 
tions make  one  township. 


MEASURES    OF   VOLUME    AND    CAPACITY. 

DEFINITIONS. 

125.  A  Volume  is  .4  magnitude  that  has  length,  breadth, 
and  thickness  or  height. 

126.  -6-  Cube  is  a  volume  or 
solid,  bounded  by  six  equal  squares, 
called  Faces  ;  the  sides  of  the 
squares  are  called  Edges  of  the 
cube.  Thus,  ABCD-E  is  a  cube, 
ABCD  is  one  of  its  faces,  AB,  BC, 
and  BE  are  edges, 

A  rectangular  volume,  or  solid,  whose  paraixelopipedon. 

edges  are  not  equal  is  a  parallelopipedon. 

127.  A  Unit  of  Volume  is  a  cube  whose  edges  are 
equal  to  the  unit  of  length.  If  each  edge  is  a  lineal  yard, 
the  cube  is  called  a  cuUc  yard. 


150 


COMPOUN^D     NUMBERS. 


128.  The  Content  of  a  volume  is  an  expression  for 
that  volume  in  terms  of  a  cubic  unit. 

Note. — It  is  shown  in  geometry  that  the  content  of  a  cube  or  of 
any  rectangular  solid  is  equal  to  the  product  of  its  three  dimensions  ; 
that  is,  the  number  of  cubic  units  in  the  volume  is  equal  to  the  num- 
ber of  units  in  its  length,  multiplied  by  the  number  of  units  in  its 
breadth,  multiplied  by  the  number  of  units  in  its  height. 

139.  A  Unit  of  Capacity  is  a  measure  having  a 
determinate  content,  or  capacity. 

I*.     CUBIC    MEASURE. 

130.  This  is  used  in  measuring  volumes  or  solids. 

TABLE. 

1728  cubic  inches  {cu.  in.)  make  1  cubic  foot cu.ft.. 

27  cubic  feet  "      1  cubic  yard cu.  yd, 

A    Cord    of 
Wood  is  a  pile  i^^^^^^^^^^^^^^^^^^^B&v 

Aft.  \yide,  4/i^.         " : .     -       mr^"^.-^^^- 

high,    and   8  ft. 

long.     A  foot  in 

length  from  such  ^P^'vi^V  eet/^^ 

a  pile  is  called  a  ^^v4^i\%l 

Cord  Foot.     A       .^  ^.^-^..NE^  cc^f  ^ 

cord  foot    is    16 

cubic  feet.  ^^^.^ 

TABLE. 

16  cubic  feet  {cu.ft)  make  1  cord  foot C.ft. 

8  cord  feet  or  128  cu.ft        "      1  cord C\ 

2°.      DRY      MEASURE. 

131.  This  is  used  in  measuring  dry  articles,  as  grain, 
fruit,  salt,  and  the  like. 

TABLE 

2  pints  {pt.)  make  1  quart qt, 

8  quarts  ''      1  peck .ph 

4  pecks  **      1  bushel hu. 


TABLES. 


151 


The  primary  unit  is  1  bushel.  The  bushel  (known  as  the  Win- 
chester bushel)  is  a  cylindrical  measure  18|^  inches  across  and  8 
inches  deep  ;  it  contains  3,150|  cubic  inches  nearly. 


133. 


3°.      LIQUID     MEASURE. 

This  is  used  in  measuring  liquids. 


TABLE. 


4    gills  (gi.)  make  1 

2    pints  ''  1 

4    quarts  "  1 

31^  gallons  "  1 

2    barrels  "  1 

2    hogsheads     '^  1 

2    pipes  "  1 

The  primary  unit  is  1  gaUon;  it  contains  231  (yubic  inches.    The 


pint pt 

quart qt. 

gallon ,gal. 

barrel bU, 

hogshead. .  .hhd, 

pipe pi. 

tun tun. 


liquid  quart  is  about  |  of  a  quart  of  dry  measure. 


ANGULAK  MEASURE  AND  LONGITUDE. 

DEFINITIONS. 

133.  A  Circle  is  a  plane  figure  bounded  by  a  curved 
line  every  point  of  which  is  equally  distant  from  a  point 
within  called  the  Centre.     The  bound- 
ing line  is  called  the  Circumference, 
and  any  part  of  this  circumference  is 
called  an  Arc  of  the  circle. 

134.  A  Diameter  is  a  line  passing 
through  the  centre  and  terminating  in 
the  circumference. 

135.  A  Radius  is  a  line  drawn  from  the  centre  to  any 

point  of  the  circumference. 

Thus,  AEDF  is  a  circle,  C  its  centre,  AD  a  diameter,  and  CE  a 
radius. 


152  COMPOUND     NUMBERS. 

\\      ANGULAR    MEASURE. 

136.  In  measuring  angles,  the  Right  Angle  (Art. 
120)  is  taken  as  the  primary  unit.  The  ninetieth  part 
of  a  right  angle  is  called  a  Degree. 

TABLE. 

60  seconds  (")  make  1  minute 

60  minutes  "      1  degree ° 

90  degrees  "      1  right  angle rt  a, 

DEFINITION    AND    EXPLANATION. 

137.  The  Longitude  of  a  place  is  the  angular  distance 
of  the  meridian  of  that  place  from  some  stmidard  meri- 
dian. It  is  measured  on  the  equator,  and  is  equal  to  the 
angle  through  which  the  earth  turns  on  its  axis  whilst  the 
sun  is  passing  from  the  meridian  of  one  place  to  that  of 
the  other.  But  the  earth  turns  through  an  angle  of  360° 
in  24  hours,  that  is,  it  turns  through  an  angle  of  15°  in 
1  hour ;  hence,  we  have  the  following  relations  between 
the  difference  of  longitude  of  two  places  and  the  difference 
of  their  local  times. 

TABLE. 

15°  of  arc  make  1  hour  of  time. 
15'   of  arc      "      1  ininute  of  time. 
15"  of  arc      "      1  second  of  time. 

Note. — All  longitudes  referred  to  in  this  book  are  reckoned  from 
the  meridian  of  Greenwich,  England. 

MISCELLANEOUS    TABLES. 

138.  The  following  miscellaneous  tables  are  often  used 
in  operating  on  compound  numbers : 


TABLES.  153 

l".      COUNTING. 

12  things  make  1  dozen doz. 

12  dozen       "      1  gross gr. 

12  gross        "      1  great  gross ff-  ff^* 

20  things      "      1  score sc, 

2\       PAPER. 

24  sheets  make  1  quire qr, 

20  quires      "      1  ream rea7n. 

2  reams      "      1  bundle bund. 

2  bundles  "      1  bale hale, 

Z\      BOOKS. 

A  book  in  which 

each  sheet  is  folded  into  2  leaves  is  a  folio. 

"        "  "        "  4  «  «  quarto,  or  4to. 

"        «  "         "  8      «  "  octavo,  or  8vo. 

"        "  "        "  12  "  "  duodecimo,  or  12ma 

"        "  "        "  16  «  "  16mo. 

"        «  u        «  24  "  "  24mo. 

«        "  «        "  32  "  "  32mo. 


THE    METRIC    SYSTEM. 

DEFINITIONS    AND    EXPLANATIONS. 

139.  The  Metric  System  is  a  system  of  weights 
and  measures  based  on  a  primary  unit  of  length  called  a 
Meter. 

This  system  was  first  adopted  by  France  and  afterwards  by  vari- 
ous other  countries.  Since  1866  its  use  has  been  permitted  in  the 
United  States,  by  act  of  Congress. 

The  meter  is  approximately  equal  to  i  o  o  o^o  o  o  o  ^^  ^^^e 
distance  from  the  equator  to  the  north  pole,  measured  on 


154  COMPOUND     NUMBERS. 

the  meridian  through  Paris.    It  is  nearly  equal  to  39.37 
inches. 

The  scales  of  all  compound  numbers  in  the  metric  sys- 
tem are  decimal 

The  names  of  units  in  the  ascending  scale  are  formed  by  prefixing* 
the  following  numerals  to  the  names  of  the  primary  units :  deca, 
ten ;  hecto,  one  hundred  ;  kilo,  one  thousand  ;  and  myria,  ten 
thousand. 

The  names  of  units  in  the  descending  scale  are  formed  by  pre- 
fixing the  following  numerals  to  the  names  of  the  primary  units : 
deei,  one  tenth ;  centif  one  hundredth  ;  and  milli^  one  thousandth. 

MEASURES    OF    LENGTH. 

140.  The  primary  unit  is  the  meter. 

TABLE. 

10  millimeters  {mm.)  make  1  centimeter  (cm.)       =        .3937  »w. 
10  centimeters  "      1  decimeter     (dm.)       —       3.937   " 

10  decimeters  "      1  meter  (m.)         =     39.37     " 

10  meters  **      1  decameter    (decam.)  =  393.7       " 

10  decameters  "      1  hectometer  {hectom.)  =    S28  ft.    1  in. 

10  hectometers  '*      1  kilometer     (kilom.)    =  S2S0  ft.  10  in. 

Lengths  are  usually  expressed  decimally  in  terms  of  some  one  of 
the  units  of  the  preceding  scale.  Small  distances  are  expressed  in 
millimeters,  ordinary  distances  in  meters,  and  long  distances  in 
kilometers. 

MEASURES    OF    SURFACE. 

141.  The  primary  unit  of  ordinary  surfaces  is  a 
Square  Meter. 

TABLE. 

100  sq.  centimeters  make  1  sq.  decimeter  =     16.6  sq.in. 
100  sq.  decimeters        "     1  sq.  meter        =  1550    "    " 

For  land  measure  the  primary  unit  is  the  are  (pron.  ar) ;  it  is  a 
square  decameter,  or  100  square  meters. 


TABLES.  155 

100  sq.  meters  make  1  are         =  119.6     sq.  yds. 
100  ares  "      1  hectare  =     2.471  acres, 

MEASURES  OF  VOLUME  AND  CAPACITY. 

143.  The  primary  unit  of  ordinary  volumes  is  the 
Stere  (pron.  stair)',  it  is  a  Cubic  Meter. 

TABLE. 

1000  cu.  centimeters  make  1  cu.  decimeter  =  61.026  cu.  in, 
1000  cu.  decimeters       "      1  cu.  meter        =  35.316  cu.ft. 

For  wood  measure  the  primary  unit  is  the  stere. 

10  decisteres  make  1  stere  {st,)  =   .2759  cords. 
10  steres  "      1  decastere  =  2.759      " 

For  measures  of  capacity  the  primary  unit  is  the  liter  (pron.  leeter). 
It  is  a  cubic  decimeter. 

TABLE. 

Dry  Measure.  Liquid  Measure. 

10  centiliters  W  make  1  deciliter   {dl.)      =    .0908  g?.=     .1057  g^. 
10  deciliters  "      1  liter  (I.)        =   .908  g^.   =    1.0567  g^. 

10  liters  "      1  decaliter  (<fec«0  =  9.08  ^««.    =    2Mn  gals. 

10  decaliters    \       "      1  hectoliter  (^c^or)=  2.8375  ?m.=  26.417  po^*. 

MEASURES    OF    WEIGHT. 

143.  The  primary  unit  of  weight  is  the  Gram.  It 
is  the  weight  of  a  cuUc  centimeter  of  distilled  water  at 
39°  Eah. 

TABLE. 

10  milligrams  {mg.)  make  1  centigram  {eg.)       =     .1543  grs. 


10  centigrams 
10  decigrams 
10  grams 
10  decagrams 
10  hectograms 


1  decigram    {dg.)       =    1.5432  " 
Igram  {g.)         =15.432     " 

1  decagram   (decag.)  =     .3527  oz.  av. 
1  hectogram  Qiectog.)  =    3.5274     " 
1  kilogram    (kilog.)    =   2.2046  lbs.  av. 


156  COMPOUI^D     NUMBERS. 

Small  weights  are  expressed  in  milligrams  and  large  ones  in 
kilograms.  The  kilogram  is  the  weight  of  a  liter  of  water  at 
39°  Fah. 

A  quintal  is  100  kilograms  or  220.46  lbs.  av.  The  French  tonneau 
(pron.  ton-no),  or  metric  ton,  is  the  weight  of  a  cubic  meter  of  water, 
or  to  2,204  lbs.  av. 

We  may  read  a  number  in  the  metric  system  in  terms 
of  all  its  units,  or  in  terms  of  any  one  of  them.  Thus, 
the  expression  27.34  meters  may  read  2  decameters,  '7  meters, 
3  decimeters,  and  4  centimeters,  but  it  is  usually  read  as  it 
is  written,  27  and  34  hundredths  of  a  meter, 

REVIE^AT     QUESTIONS. 

(105*)  What  is  a  denominate  number?  When  are  numbers  of 
the  same  denomination  ?  Of  different  denominations  ?  ( 1 OO.)  What 
is  a  compound  number  ?  When  are  numbers  of  the  same  kind  ? 
Of  different  kinds ?  (107.)  What  is  a  scale?  Give  the  scale  of 
English  currency?  Of  United  States  currency?  (111.)  What  is 
the  primary  unit  of  weight  for  Great  Britain  and  the  United  States  ? 
How  was  it  established  ?  How  many  grains  in  a  pound  Troy  ? 
How  many  in  a  pound  avoirdupois?  (114.)  What  is  time? 
Explanation  of  the  lengths  of  years  and  of  their  distribution  ? 
What  years  are  leap  years  ?  What  years  are  common  years  ?  Name 
the  months  and  the  number  of  days  in  each.  (115.)  Define  mag- 
nitude. (116.)  Define  a  line.  What  is  the  length  of  a  line? 
(1 19.)  Define  a  surface.  (120.)  Define  an  angle.  A  right  angle. 
(121.)  Define  a  square.  (122.)  Define  a  unit  of  surface.  Area. 
(125.)  Define  a  volume.  (126.)  Define  a  circle.  (127.)  Define 
a  unit  of  volume.  Content  of  a  volume.  Unit  of  capacity.  (1 80.) 
Wliat  is  a  cord  of  wood  ?  What  is  a  cord  foot?  (133.)  Define  a 
circle.  Circumference.  Arc.  (134.)  Define  a  diameter.  (135.) 
Define  a  radius.  (136.)  What  is  a  degree  ?  (137.)  Define  longi- 
tude. What  is  the  relation  between  longitude  and  time,  and  how 
was  it  established ?  (139.)  What  is  the  metric  system  ?  A  meter? 
How  are  the  names  of  metric  numbers  formed?  (141.)  What  is 
an  are?  For  what  purpose  is  it  used?  (142.)  What  a  stere? 
A  liter?    (143.)  What  is  a  gram  ? 


EEDUCTIOK.  167 

II.      REDUCTION. 

DEFINITIONS. 

144.  Reduction  of  denominate  numbers  is  the 

operation  of  changing  a  number  from  one  denomination 

to  another,  without  altering  its  value. 

A  number  may  be  changed  from  a  higher  to  a  lower  denomination, 
or  from  a  lower  to  a  higher  denomination  ;  in  the  former  case  the 
operation  is  called  reduction  descending,  in  the  latter  it  is  called 
reduction  ascending, 

REDUCTION    DESCENDING. 

145.  Reduction  descending  is  the  operation  of 
changing  a  number  from  a  higher  to  a  lower  denom- 
ination. 

MENTAL      EXERCISES. 

1.  How  many  pence  are  there  in  1  shilling  9  How  many 
in  bs,9  in  145..^  In  l/oo^,  how  many  inches  ?  in  ^ft.9 
in  IQftJ  In  1  hour,  how  many  minutes?  in  3  hrs.f 
in  9  hrs.  9  in  15  hrs,  ?  in  18  hrs.  ? 

2.  How  many  ounces  in  1  pound  avoirdupois  ?  in  5  lbs.  9 
in  8  lbs.  f  in  9  lbs.  9  In  1  pound  sterling,  how  many  shil- 
lings 9  in  £6  ?  in  £11  ?  in  £14  ?  In  5  yards^  how  many 
inches  9  in  12  yds.  9  in  14  yds.  8  in.  9 

Explanation. — In  14 yds.  there  are  14  x  12,  or  168 inches;  hence, 
in  14  yds.  5  in.  there  are  168  +  5,  or  173  iiiches. 

3.  In  7  shillings,  how  many  pence  9  in  ^s.  6d.  ?  in 
95.  9d.  f  In  3  pecks,  how  many  quarts  9  in  2pks.  7  qts.  9 
in  3  2^ks.  5  qts.  f  In  4  hundredweight,  how  many  quar- 
ters ?  in  9  ciut.  3  qrs.  9  in  12  civt.  2  qrs.  9  In  7  meters,  how 
many  decimeters  9  in  Sm.  '7  dm.  9  in  S.ll  m.9  in  11.4  m.  9 
In  8  dollars,  how  many  cents  9  in  19.50  ?  in  114.15  ? 


158  COMPOUND     NUMBERS. 


100  COMPOUND     NUMBERS. 

4.  In  4^(Zs.,  how  m'dn^feet?  How  many  inches?  In 
6  yds.,  how  many  incliesl  in  5  yds,  4  in.  ?  in  7  ?/f/s.  9  m  ? 
How  many  i^c/^es  in  6  ^/^s.  2//5.  3  ^■?^.  ? 

Explanation.— In  6  yds,  2  ft.  there  are  30//5.,  and  in  20/if.  9  »;i. 
there  are  249  inches. 

5.  In  £2  10s.,  how  many  shillings  9  how  many  pence  ? 
In  £3  45.,  how  many  pence  f  How  many  pints  in  7  qts.  ? 
in  6  qts.  Ipt.  9  in  2jt?^5.  2  qts.  ?  in  3  ^^5.  4  (7^5.  1^^.  .^  In 
4  meters,  how  many  decimeters  ?  how  many  centimeters  9 
how  many  millimeters?  In  14.15,  how  many  mills?  in 
$5.07?  in  $4,723? 

OPERATION    OF    REDUCTION    DESCENDING. 

146.  Let  it  be  required  to  reduce  £7  4s.  Sd.  to  pence : 

Explanation. — The  given  number 
contains  7  units,  each  equal  to  £1, 
and  from  the  scale  (Art.  109.)  we  see 
that  each  pound  contains  20s. ;  mul- 
tiplying 7  by  20  and  adding  4,  we 
have  144,  that  is,  £7  4s.  is  equal  to 
144s. ;  but  from  the  scale  we  see  that 
each  shilling  is  equal  to  12d.  ;  mul- 
tiplying 144  by  12  and  adding  3  we  have  1781 ;  that  is,  the  given 
number  is  equal  to  1731c?. 

Since  all  similar  cases  may  be  treated  in  the  same  manner,  we 
have  the  following 

RULE. 

/.  Multiply  the  units  of  the  highest  denomina- 
tion by  the  ?iumber  of  the  scale  that  connects 
this  denomination  ivith  the  one  next  lower  and 
add  the  units  of  the  latter  denomination  to  the 
product. 

II.   Multiply    this    result    by    the    number    that 


OPERATION. 

£7 

4.S. 

Bd. 

20 

1445 

12 

1731^. 

Ans. 

REDUCTION.  159 

connects  it  with  the  next  lower  denomination 
and  add  the  units  of  that  denomination  to  the 
product. 

III.  Continue  this  operation  till  the  required 
denomination  is  reached. 

EXAMPLES. 

1.  How  many  farthings  are  there  in  £4  lis.  3d  2  far  •'I 

2.  How  many  grains  in  3  lbs.  Boz.  6divts.  ^grs.f 

3.  How  many  minutes  in  3  wks.  5  da.  5  hrs.9 

4.  How  many  yards  in  3  mi.  28  rds.  2  yds.  ? 

5.  How  many  square  feet  in  40  square  rods  ? 

6.  How  many  pints  in  8  hu.  Spks.  2  qts.  Ipf.  ? 

7.  How  many  seconds  in  4:ivlcs.  dda.  l'7hrs.? 

8.  In  £31  85.  9}d,  how  many  farthings  ? 

9.  How  many  inches  in  6  rds.  4  yds.  2ft.  9  in.  ? 

10.  In  17.5  square  chains,  how  many  square  links  ? 

11.  In  25  cords  of  wood,  how  many  cord  feet  and  how 
many  cubic  feet  ?    In  172  C.  ?    In  115  C.  8  G.ft.  ? 

12.  How  many  quarts  are  there  in  20  hu.  Zphs,  ? 

13.  Reduce  £|-  to  shillings  and  pence. 

Explanation. — Multiplying  by  20  to  reduce  pounds  to  shillings, 
we  have  £|  =  -^-s.  =  16|«. ;  multiplying  |.9,  by  12  to  reduce  shillings 
to  pence,  we  have  f  s.  =  ^^d.  =  Sd. :  hence,  £f  =  16s.  Sd.  Ans. 

In  like  manner  we  may  reduce  any  denominate  fraction  to  in- 
tegral units  of  a  lower  denomination  ;  hence  the  following 

RULE. 
Multiply  the  fraction  hy  the  number  that  con- 
nects it  with  the  next  lower  denomination ;  then 
multiply  the  fractional  part  of  the  result  hy  the 
number  that  connects  it  with  the  next  lower  denom- 
ination, and  so  on. 


160  COMPOUIfD     NUMBEKS. 

14.  Reduce  .H 14: yds.  to  feet  and  inches. 

Solution.— Multiplying  by  3,  we  have  .IHyds.  =  2.142  ft;  mul- 
tiplying .143/^.  by  12,  we  have  .142  ft.  =  1.704 in. ;  hence,  .714yds. 
=  2 ft.  1.704  in.  Ans. 

Reduce  the  following  denominate  fractions  to  integral 
units  of  different  denominations : 

15.  £.875.  24.  %^mi.  33.  £|4. 

16.  .^'dlblhs.  Tr.        25.  imfur,  34.  4^hu, 
V^.  .11^70 tons.           m.d^lhu.              35.2.56  21 

18.  .46sq.yds.  27.  43^  yds.  36.  3.567  ^«. 

19.  £2|.  28.  17% yds.  37.  .oUmi. 

20.  d^da.  29.  4.562  M^5.  38.  1.895. 

21.  141}  w^s.  30.  3J^°.  39.  3.46.4. 

22.  4^  T.  31.  12M62.  40.  .467 gals. 

23.  3ifm  ^.  32.  .436  T.  41.  ^f^  mi. 

REDUCTION     ASCENDING. 

147.  Reduction  Ascending  is  the  operation  of 
changing  a  number  from  a  lower  to  a  higher  denom- 
ination. 

MENTAL     EXERCISES. 

1.  In  12  inches  how  many  feet  9  In  96  in.  how  many  ft.  9 
in  144  in.f  in  204  in.9  In  120  seconds  how  many  minutes  9 
in  dQOsec.f  In  16  §'^5.  how  msLnj  pJcs.f  in  dQqts.P  How 
many  minutes  in  240"  ?  in  540"  ?  How  many  degrees  in 
300'  ?  in  540'  ?    Reduce  600'  to  (^^^rees. 

2.  In  64  02;.  «?;.,  how  msinj  pounds  f  In  100  o^:.  «v.,  how 
many  pounds  and  how  many  ounces  remain  ?  Reduce 
110  oz.  av.  to  pounds.  Ans.  6  lbs.  14:  oz.  In  219  m.  how 
msLHj  feet  and  inches  9  Reduce  154  iw,  to  a  higher  denom- 
ination.   Ans.  12  ft.  10  in. 


REDUCTION-.  161 

3.  In  64:pks,  how  many  iu.f  In  74  ^^5.  how  many  bu, 
and  how  many  pks.  remain  ?  In  69pks.  how  many  bu, 
and pks.f  Eeduce  VZH qts.  to  pks.  Eeduce  U7d.  to  sMl- 
lings  and  pence.     Eeduce  121i?iches  to  feet  and  inches, 

4.  In  1456?.  how  many  shillings  and  how  manj pence? 
In  163 inches  how  manjfeet?  Ans.  13ft.  7 in.  In  ^3 ft. 
how  many  2/^5.  .^  Ans.  4:  yds.  1ft.  How  many  yds.  ft. 
and  ^/^.  in  163  in.  f  Ans.  4:  yds.  1ft.  7  in.  Eeduce  213  in. 
to  higher  denominations.  Express  265 pts.  in  higher  de- 
nominations.    Eeduce  2S0d,  to  higher  denominations. 

OPERATION  OF  REDUCTION  ASCENDING. 

148.  Let  it  be  required  to  reduce  1,731  pence  to  pounds, 
shillings,  and  pence : 

Explanation.— We  see  from   tte  operation. 

scale,  (Art.    109),   that    13    connects  12)  173 Id 

pence  with  shillings ;    dividing  1,731  20)144s.        3d. 

by  12,  we  find  a  quotient  144  with  a  — ^r^ , 

remainder  3  ;  hence  the  given  number  •  •  •    5. 

is  equal  to   144*.  Sd. ;  in  like  manner  £7  45.  3d.     Ans. 

dividing  144  by  20,  we  find  a  quotient 

7  and  a  remainder  4,  that  is,  144s.  is  equal  to  £7  4s. ;  hence,  the 
given  number  is  equal  to  £7  4s.  Sd. 

In  like  manner  we  may  treat  all  similar  cases  ;  hence,  the  fol- 
lowing 

RULE. 

/.  Divide  the  units  of  the  given  denomination  hy 
the  number  of  the  scale  that  connects  this  denom- 
ination with  the  one  next  higher ;  the  remainder 
mill  he  units  of  the  same  denomination  as  the  divi- 
dend. 

II.  Divide  the  quotient  hy  the  numher  that  con- 
nects it  with  the  next  higher  denomination ;  the 
6 


162  COMPOUND     NUMBERS. 

remainder  will  he  units  of  the  same  denomination 
as  the  new  dividend. 

III.  Continue  the  operation  till  the  required  der 
nomination  is  reached. 

EXAMPLES. 

1.  Eeduce  15,732  grains  to  pounds  Troy. 

2.  Eeduce  525,960  minutes  to  weeks. 

Proof. — Reduction  descending  is  proved  by  reduction  ascending, 
and  the  reverse ;  thus,  the  preceding  example  is  proved  by  reducing 
52  wks.  1  da.  6  hrs.  to  minutes ;  this  gives  525^960  min. 

3.  Express  49,180 ^mms  m pounds,  Apothecaries'  weight. 

4.  Eeduce  d,392  grains  to  ounces,  Apothecaries'  weight. 

5.  Eeduce  2,945  inches  to  higher  denominations. 

Explanation. — In  this  case  we  reduce  3,945  inches  to  half  yards; 
then  we  reduce  the  half  yards  to  rods,  (Art.  117).  Dividing  2,945 
inches  by  18,  because  there  are  18  inches  in  a  half  yard,  we  find  163 
half  yards  and  a  remainder  of  11  inches  ;  dividing  163  by  11,  because 
there  are  11  half  yards  in  a  rod,  we  find  14  rods  with  a  remainder 
equal  to  9  half  yards,  or  to  4  yards  and  18  inches ;  adding  the  18 
inches  to  the  first  remainder,  11  inches,  we  find  29  inches,  or  3/if.  5  in. 
Hence,  2,945  in.  =  14rc?5.  4:  yds.  %ft.  5  in. 

Eeduce  Eeduce 

6.  1,365  in.  to  rods,  14.  16,411  oz,  to  cwts. 

7.  272,668  in.  to  miles.         15.  311,375  sec.  to  days. 

8.  88,435  ^?^.  to  miles,  16.  31,463  mm.  to  weeks. 
,  9.  873  0;?.  av.  to  quarters.     17.  21,118"  to 

10.  7,634^t.  to  gallons.  18.  1,114'  to 

11.  8,372 /«r.  to  pounds,        19.  4:,64:3mm.  to  meters. 

12.  14,311  ^r.  to  lbs.  Tr.        20.  13,362.9^.  in.  to  sq.  yds. 

13.  4,771 3  to  Ws.  21.  1,211,312  sq. li.  to  acres. 
^2.  Eeduce  75.  ^d.  to  a  fractional  part  of  £1. 


REDUCTION.  163 

Explanation. — Reducing  7s.  Qd.  to  pence,  we  have  7s.  6d.  =  90d; 
reducing  the  given  unit  £1  to  pence,  we  have,  £1  =  240c?. :  hence, 
7s.  Od.  =  £ff  =  £|.  Ans.  In  like  manner  all  compound  numbers 
may  be  expressed  in  fractional  parts  of  a  higher  integral  unit ;  hence 
the  following 

RULE. 
Reduce  the  compound  nurriber  to  its  lowest  denom- 
ination ;    also  reduce  the  given  unit  to  the  same 
denomination ;  then  divide  the  former  by  the  latter. 

23.  What  part  of  a  tun  is  3  hhds.  31  gals.  2  qts.f 

24.  What  part  of  a  hogshead  is  3  gals.  2  qts.  9 

25.  What  part  of  a  mile  is  116  rds.  2  yds.  9 

26.  What  part  of  a  right  angle  is  3°  15'  12"  ? 

27.  What  part  of  a  gallon  is  2  qts.  Ipt.  Igi.  9 

28.  What  part  of  a  cord  is  4  C.  ft.  7  cu.ft.  9 

29.  What  part  of  a  meter  is  714  mm.  9 

30.  What  decimal  of  a  ton  is  15  cwt,  3  qrs.  2|-  lbs.  9 
Explanation. — For  convenience   the  operation. 

units  of  the  several  denominations   are  25  |    2.5  Ibs. 

written  in   a  vertical   column,  the  least  ~7~i     oT/y^o 

denomination  at  the  top,  and  the  num-  ^'^^^ 

bers  of  the  corresponding  scale  are  writ-  /30  |  Ib.llb  CWt. 

ten  on  the  left  from  the  top  downward.  .78875  T. 

Dividing  2^lbs.,  or  its  equal  2.5  lbs.,  by  25, 

to  reduce  it  to  quarters,  we  have  .1  gr.,  which  being  annexed  to 
3  qrs.  gives  3.1  qrs. ;  dividing  3.1  qrs.  by  4,  to  reduce  it  to  hundreds, 
we  have  .775  cwt.,  which  being  annexed  to  15  cwt.  gives  15.775  cwt.; 
dividing  this  by  20  to  reduce  it  to  tons,  we  have  .78875  T.,  which  is 
the  answer. 

In  like  manner  any  compound  number  may  be  reduced  to  a  deci- 
mal of  a  higher  denomination  ;  hence  the  following 

RULE. 
Divide  the  units  of  the  lowest  denomination  by 
the  number  of  the   scale   that  connects   this    de- 


^ 


164  COMPOUND     NUMBERS. 

nomination  with  the  next  higher  one,  (Art.  107 j, 
and  annex  the  quotient  to  the  units  of  that  de- 
nomination ;  then  divide  this  result  by  the  number 
that  connects  it  with  the  next  higher  denomina- 
^  tion  and  annex  the  quotient  to  the  units  of  that 
denomination;  and  so  on,  till  the  required  denom- 
ination is  reached. 

31.  In  5  hu.  3pks.  6  qts.,  how  many  pechs  9 

Explanation. — We  first  reduce  5  bu.  3  pks.  to  pks.,  which  gives 
2dpks.;  we  then  reduce  6qts.  to  a  decimal  of  &pk.,  which  gives 
!75  pks. ;  hence,  5bu.  3 pks.  Qqts.  —  23.76 pks.,  Ans. 

Reduce  the  following  to  the  units  indicated : 

32.  £14  175.  U,  to  pounds.  39.  2  T.  ^cwt.  3qrs.  to  lbs. 

33.  7  mi.  281 7'ds.  to  miles.  40.  8  bu.  Spies.  6  qls.  to  pks. 

34.  2cwt.  Iqr.  12ybs.  to  tons.  41.  £5  45.  9d.  to  shillings. 

35.  3J1  to  lbs.  (Apoth.)  42.  Sbbls.  14:^  gals,  to  gals. 

36.  2^pts.  to  gallo7is.  43.  47  rds.  4|  yds.  to  yds. 

37.  21  in.  to  yds.  44.  47°  35'  42"  to  minutes. 

38.  2  /Js.  4  02;.  5  diuts.  to  02?.  45.  4/??.  Sdm.  to  decameters. 

46.  What  part  of  £3  IO5.  6^.  is  £1  155.  M.  ? 

Explanation. — Here  we  reduce  both  numbers  to  perice,  the 
lowest  denomination  named  in  either  ;  we  then  perform  the  operation 
indicated  in  tlie  question.  Thus,  £3  10s.  Qd,  =  846(^.,  and  £1  15s.  4d. 
=  424(^. ;  hence  the  required  answer  is  |ff  or  |^. 

47.  What  part  of  36°  15'  20'  is  11°  13'  20"  ? 

48.  What  part  of  4:  gals.  3  qts.  is  2  gals.  3  qts.  ? 

49.  What  part  oi2T.  6  cwt.  20  lbs.  is  7  cwt.  10  lbs.  9 

50.  What  part  of  £|  is  1 J5. 9 

51.  What  part  of  -J  mi.  is  4J  rds.  9 

52.  What  part  of  3  bu.  3  pks.  is  2  bu.  Ipk.  9 


REDUCTIOlif.  166 

MISCELLANEOUS    EXAMPLES    IN    REDUCTION. 

1.  How  many  grains  in  111),  11  oz.  15  clivts.  9 

2.  In  97,397  grains  of  gold  how  many  pounds  Troy? 

3.  In  24  tons  17  cwL  3  qr.  how  many  pounds? 

4.  In  13 6 J  bushels  how  many  quarts  ? 

5.  How  many  miles  in  1,571,328  inches  ? 

6.  In  2,624  cubic  feet  of  wood,  how  many  cords  ? 

7.  Reduce  18,545,435  sec,  to  days.     To  weeks. 

8.  What  is  the  value  of  |  of  Is.  ?     Of  i  of  Qs.  ? 

Note.— Metric  weights  and  measures  may  be  converted  into 
English  weights  and  measures  by  the  tables  of  equivalents  given 
in  Arts.  140-143. 

9.  How  many  inches  in  7  meters  ?    In  13  meters  ? 

10.  How  many  meters  in  605  inches  ?    In  319  in.  9 

11.  How  many  miles  in  12  kilometers? 

12.  How  many  kilometers  in  15  miles  ?    In  7 J  7m.  9 

13.  How  many  quarts  (liquid  measure)  in  41  liters  ? 

14.  How  many  gallons  in  4  hectoliters  ? 

15.  How  many  liters  in  31|-  gallons  ?    In  74  galls,  9 

16.  How  many  ounces  in  711  grammes  ?    In  51  g.  9 

17.  How  many  pounds  in  74  kilogrammes  ? 

18.  Reduce  510  kilogrammes  to  pounds. 

19.  How  many  cords  in  15  steres  of  wood? 

20.  Reduce  17J  steres  of  wood  to  cords.     To  cu.  ft, 

21.  Reduce  175.  9|^.  to  a  decimal  of  a  pound. 

22.  What  part  of  £1  2s.  6d.  to  Is.  Ud.  9 

23.  Reduce  1  lb.  9  oz.  15  gr.  to  pounds.     To  ounces. 

24.  Reduce  3  da.  14  hrs.  25  min.  to  weeks. 

25.  What  part  of  a  barrel  is  8  gals.  2  qts  9 

26.  What  part  of  a  mile  is  74  rds,  5  yds.  9 


166  COMPOUND     NUMBEKS. 

27.  What  part  of  a  cord  is  3f  cord  feet? 

28.  What  part  of  a  pound  sterling  is  5s.  9d.  f 

29.  Eeduce  £.187  to  lower  denominations. 

30.  Eeduce  .574  miles  to  lower  denominations. 

REVIE^A/'    QUESTIONS. 

(14:4.)  What  is  reduction  of  denominate  numbers?  How  many 
kinds  of  reduction?  (145.)  What  is  reduction  descending? 
(146.)  Rule  for  reduction  descending?  Rule  for  reducing  de- 
nominate fractions  to  integral  units?  (147.)  What  is  reduction 
ascending?  (148.)  Rule  for  reduction  ascending?  Rule  for  re- 
ducing compound  numbers  to  any  unit  of  the  kind  named?  Rule 
for  finding  the  part  that  one  compound  number  is  of  a  similar 
number  ? 


III.    ADDITION    OF    COMPOUND 
NUMBERS. 

DEFINITION. 

149.  Addition  of  Compound  Numbers  is  the' 
operation  of  finding  the  sum  of  two  or  more  compound 
numbers  of  the  same  kind. 

MENTAL    EXERCISES. 

1.  What  is  the  sum  of  9  inches  and  11  inches?  Of  13 
inches  and  .22  inches?  Of  ^ft.  and  It  ft.  ?  What  is  the 
sum  of  IM.  and  16f7.  ?    Of  10^.,  Id.,  and  IM.  ? 

2.  What  is  the  sum  of  7  ft.  and  8  ft.  ?  How  many  yards 
in  7  ft.  ?  In  8  ft.  f  In  the  sum  of  7  ft.  and  8  ft.  ?  What 
then  is  the  sum  of  2  yds.  1  ft.  and  2  yds.  2  ft.?  What 
is  the  sum  of  11  qts,  and  15  qts.  ?  How  many  gallons  in 
11  qts.  ?  In  15  qts.  ?  In  11  qts.  +  15  qts.  ?  What  then 
is  the  sum  of  2  gals.  3  qts.,  and  3  gals.  3  qts. 


ADDITION.  167 

3.  What  is  the  sum  of  3  gals,  and  5  gals,  f    Of  2  qts. 

and  3  qts.  9    Of  3  gals.  2  qts.,  and  5  gals.  3  qts.  9    What 

is  the  sum  of  35.  8d,  and  55.  M.  ^    Of  3  yds.  2  ft.,  5  yds. 

lfL,2^iidi1  yds.  2  ft  J 

Note. — The  principles  used  in  the  addition  of  compound  numbers 
are  the  same  as  those  used  in  the  addition  of  dmple  numbers. 

OPERATION    OF    ADDITION    OF    COMPOUND    NUMBERS. 

150.  Let  it  be  required  to  find  the  sum  of  £7  45.  M., 

£11  95.  M.,  and  £14  125.  M. : 

Explanation.  —We  write  the   num-  opebation. 

bers  so  that  units  of  the  same  denom-  £          g^        d^ 

ination  shall  stand  in  the  same  column.  7          4          S 

The  sum  of  the  numbers  in  the  first  ^^           ^ 

column  is  20(Z.,  or  \s.  M.  (Art.  147) ;  ^ 

setting   down  8(f.,   we   carry  forward  14       \Z         9 


\s.,  and  add  it  to  the  second  column.  £33  Qs.      8d. 

The  sum  of  the  numbers  in  the  second 

column,  thus  increased,  is  26^.,  or  £1,  6*.;  setting  down  6s.,  we 
carry  forward  £1  to  the  next  column,  which  then  amounts  to  £33. 
The  required  sum  is  therefore  £33  6*.  Sd. 

In  like  manner  we  may  treat  all  similar  cases  ;  hence,  the  fol- 
lowing 

RULE. 

I.  Write  the  nunibers  so  that  units  of  the  same 
denomination  shall  stand  in  the  same  column. 

II.  Add  the  units  of  the  lowest  denomination 
and  divide  their  sum  by  the  number  of  the  scale 
that  connects  this  denomination  urtth  the  next 
higher  one ;  set  down  the  remainder  and  carry  the 
quotient  to  the  next  column. 

III.  Add  the  units  of  the  second  column  thus 
increased,  and  proceed  as  before,  continuing  the 
operation  till  all  the  columns  have  been  added. 


168  COMPOUND     NUMBEBS. 

EXAM  P  LES. 

Add  the  following  compound  numbers : 

(1.)  ,  (%)  (3.) 

£       s.  d.          cwt.  qrs,  lbs.  3  3  grs. 

17      13  11            2  3  27  3      1  17 

13      10  2            1  1  17  2      3  19 

10      17  3            4  2  26  6      1  10 


£42        Is.      4:d.         9ctvt.0qrs.20ll?s.     12  3    13     Qgrs. 

Proof. — The  method  of  proof  is  the  same  as  in  addition  of  sim- 
ple numbers. 


(4.) 

(5.) 

(6) 

bu. 

pics. 

qts. 

pks. 

qts. 

pis. 

gals. 

3'i^5. 

pts. 

17 

2 

5 

3 

7 

1 

2 

3 

1 

34 

2 

7 

2 

6 

1 

5 

1 

1 

13 

3 

6 

0 

4 

0 

7 

2 

0 

16 

3 

4 

3 

5 

1 

11 

3 

1 

(7.) 

(8.) 

(9.) 

yds. 

ft- 

in. 

A. 

sq.  ck 

.  sq.  li. 

^«. 

hrs. 

min. 

4 

2 

11 

5 

4 

300 

4 

14 

30 

3 

1 

8 

2 

7 

185 

3 

12 

15 

1 

1 

9 

9 

4 

1230 

5 

4 

20 

6 

2 

1 

1 

8 

211 

6 

16 

18 

(10.) 

(11.) 

(12.) 

£ 

s. 

d. 

lbs. 

02;. 

dwts. 

mi. 

rds. 

yds. 

18 

4 

9 

96 

10 

19 

2 

120 

4 

7 

11 

6 

4 

6 

16 

8 

72 

3 

9 

18 

9 

5 

10 

15 

2 

112 

4 

ADDITION.  169 

13.  Find  the  sum  of  IKS.  6d. ;  £3  5s.  Sd. ;  £25  lis. 
iOid.j  £12  Os.  Sd.  J  and  £50  4s.  ^d, 

14.  Add  S7b2i.  IpL  Sqts,;  4:1  hi.  2 pics.  6qts.;  34:  bu, 
Ipk.  3  qts.j  and  43  bu.  3pks.  1  qt. 

15.  It  took  a  carpenter  3  days  to  build  a  fence :  the  first 
day  he  built  4  rds.  4  yds.  1ft.  4  in.  ;  the  second  day  3  rds. 
2  yds.  2  ft.  9  in. ;  the  third  day  4  rds.  3  yds.  1ft.  7  in. 
What  was  the  length  of  the  fence  ? 

Reducing  to  half  yards  and  inches  (Art.  117.) 

r(?s.    ?/6?s.    /if.      in.  rds.  hf.yds.  in, 

4414  =         4            8  16 

3229  =         3            5  15 

4317  =        4            7  1 


Ans.  12  rds.  10  4/*.  ^<?5. 14  in. 
=       12  r6?s.  5  yds.  1ft.  2  m. 

If  some  of  the  numbers  are  fractional,  they  may  all  be  reduced 
to  decimals  of  the  same  unit  and  then  added  by  the  rule  in 
Art.  92. 

16.  Find  the  sum  of  3s.  6d.j  £J;  and  £.875. 

17.  Add  3  da.  16  hrs. ;  f  of  1  da. ;  and  .632  da. 

18.  Add  £13  14s.  Sd. ;  £1  7s.  2id. ;  £3  13s.  9id. ; 
£12  12s.  3|^. ;  and  £17  14s.  Sd. 

19.  Add  4  02;.  15  dtvts.  12  grs.  ;  S  oz.  10  dwts.  lH  grs. ; 
lloz.  14 dwts.  16 grs.  ;  and  10  oz.  IS  dwts.  29 grs. 

20.  Add  10?  7  3  23  14 ^rs.;  21  43  13  18 ^rs.; 
11  33  13  15^rs.;  and  3  3  23  ll^rs. 

21.  Add  7  cwt.  3  qrs.  14  lbs. ;  4  cwt.  2  qrs.  20  lbs. ;  1  cwt. 
1  qr.  10  lbs. ;  3  civt.  1  qr.  17  lbs. ;  5  cwt.  1  qr.  8  lbs. ;  and 
7  cwt.  3  qrs.  10  lbs. 


170  COMPOUND     KUMBERS. 

22.  Add  7  wks,  dda.  11  hrs. ;  5  whs.  4  da.  19  hrs. ; 
11  wy(;s.  2  6?a.  13  ^rs.  ;  1  tvk,  6  <?a.  17  hrs.  ;  and  12  wA;-?. 
4  c?«.  3  ^rs. 

23.  Add  2  mi.  ISO  rds.  6  yds,  2ft.;  dmi.  112  rds.  3  yds. 
1ft.  J  8  mi.  300  rds.  b^yds.;  and  11  mi.  18  r6?5.  3  yds.  2  ft. 

24.  Add  '^^yds.;  n^yds. ;  l^yds.;  \\\^yds. ;  and 
16^2^^^. 

25.  K^^Yllu.^'plcs^^qts.;  112 lu.  2 phs.  "i qts,;  ^Ihu. 

26.  29.65  T  +  87.25  civt.  +  19  ?^>s.  =  9 

27.  1|  cwi^.  +  17.25  Ihs.  +  49  Z^>5.  =  F 

28.  £32.5  +  17.55.  +  37^8.  =  f 

29.  47.5  da.  +  34.2  ^a.  +  "^f  da.  —  f 

PRACTICAL     PROBLEMS. 

1.  A  merchant  sent  off  the  following  quantities  of  but- 
ter :  47  cwt.  2  qrs.  7  Ihs. ;   38  mvt.  3  qrs.  8  Ihs. ;  and  16  cwt. 

2  qrs.  20  Ihs. ;  how  much  did  he  send  off  in  all  ? 

2.  A  silversmith  has  3  parcels  of  silver :  the  first  con- 
tains 7  Ihs.  Soz.  16  dwts. ;  the  second  contains  9  Ihs.  7  oz. 

3  dwts.  ;  and  the  third  contains  4  ZJs.  1  dwt. ;  how  much 
has  he  in  all  ? 

3.  A  merchant  sells  cloth  as  follows:  to  A.,  16^  yds.; 
to  B.,  90^  yds. ;  and  to  C,  19033g-  yds. ;  how  much  does 
he  sell  to  all  ? 

4.  A  man  has  three  farms  :  the  first  contains  120  A. 
74 sq.  rds. ;  the  second  contains  75 ^.  46 sq.  rds.;  and  the 
third  contains  97  A.  46  sq.  rds. ;  how  much  do  they  all 
contain  ? 

5.  B.  aged  14  yrs.  6  mos.  goes  out  to  service  ;  he  lives  at 
one  place  1  yr.  9  mos.,  at  another  place  2  yr.^.  5  7nos.,  and  at 
a  third  place  3  yrs.  9  mos. ;  how  old  is  he  then  ? 


ADDITION.  171 

6.  A  man  spent  111.26  francs  for  a  vest,  62.17 /r.  for  a 
coat,  and  38.29 /r.  for  a  pair  of  boots;  how  many  dollars 
did  he  spend  in  all  ? 

7.  A  man  sold  4  cheeses:  the  first  weighed  9.25  Icilog., 
the  second  10.14  hilog.,  the  third  11.16  kilog.,  and  the  fourth 
10.77  hilog. ;  how  many  pounds  did  they  weigh  ? 

8.  In  a  farm  there  are  5  fields :  the  first  contains  18  A. 
8  sq.  ch.,  the  second  12  J.  3  sq.  cli.,  the  third  9  ^.  4  sq.  ch., 
the  fourth  11^.,  and  the  fifth  16  J[.  2sq.ch.j  what  is  the 
content  of  the  farm  ? 

9.  How  many  yards  in  4  pieces  of  cloth  measuring  as 
follows:  SOi tjds.,  21!^ yds.,  39-^ yds.,  and  S'^iyds.? 

10.  A  man  bought  3  loads  of  wood :  the  first  contained 
1  C.  17  cu.ft.f  the  second  1  C.  115  cu.ft.,  and  the  third 
1  0.2  C.ft. ;  how  much  wood  did  he  buy  ? 

REVIE^A^      QUESTIONS. 

(149.)  What  is  addition  of  compound  numbers?  (150.)  What 
is  the  rule  for  addition  of  compound  numbers  ? 


IV.  SUBTRACTION  OF  COMPOUND 
NUMBERS. 

DEFINITION. 

151.  Subtraction  of  Compound  Numbers  is  the 
operation  of  finding  the  Difference  of  two  numbers  of 
the  same  kind. 

MENTAL     EXERCISES. 

1.  What  is  the  difference  of  l^qts.  and  9qts.9  of  7phs. 
and  4:phs.  ?  of  18  yds.  and  11  yds.  ?  of  30  ds.  and  17  cts.  9 
of  25  rds.  and  14  rds.  f  of  16  mi.  and  11  mi.  f 


172  COMPOUND     NUMBERS. 

2.  What  is  the  difference  of  Sid.  and  IBcl  9  How  many 
shillings  in  31^..?  in  16d.f  In  the  difference  between 
did.  and  16d.f  What  is  the  difference  between  2^.  7d. 
and  Is.  4:d.  9     2s.  7^.  —  Is.  4:d.  =  9 

3.  How  many  yards  in  26  ft.  9  in  16//.  P    What  then 

is  the   difference   between   8  yds.  2  ft.  and  h  yds.  1ft.  9 

between  %yds.  1ft.  and  Q  yds.  2  ft.  9  between  1  gals.  Iqt. 

and  5  gals.  3  qts.  9 

Note.— The  principles  used  in  subtraction  of  compound  numbers 
are  the  same  as  those  used  in  mbtraction  of  simple  numbers. 

OPERATION    OF    SUBTRACTION    OF    COMPOUND    NUMBERS. 

152,  Let  it  be  required  to  find  the  difference  between 
£9  4s.  M.  and  £2  ISs.  6d.  : 

Explanation.— We  write  the  subtra-  operation. 

hend  under  the  minuend  so  that  units  £         S.  d. 

of  the   same  denomination   shall  stand  9  4  3 

in  the  same  column.     Beginning  at  the 
lowest  denomination,   we   see   that  6d.  1"  "  . 


cannot  be  taken  from  M. ;  we  therefore  £q  k  <,        q^ 

add  12d.  to  Sd.,  which  gives  15^.,  and 

then  subtract  6d.  from  the  sum  ;  the  remainder,  9d.,  we  set  down, 
and  to  compensate  for  the  12d.  added  to  the  minuend,  we  add  its 
equal.  Is.,  to  the  next  column  of  the  minuend.  The  sum,  19s., 
being  greater  than  4«.,  we  add  20^.  to  the  latter  and  subtract  19s. 
from  the  sum  ;  the  remainder,  5s.,  we  set  down  and  as  before  carry 
forward  20s.,  or  its  equal  £1,  and  add  it  to  the  minuend,  giving 
£3  ;  this  taken  from  £9  leaves  £6  :  hence,  the  required  remainder 
is  £6  5s.  9d. 

In  like  manner  we  may  treat  all  similar  cases ;  hence,  the  fol- 
lowing 

RULE. 

/.   Write  the  subtrahend  under  the  minuend  so 

that  units  of  the  same  denomination  shall  stand 

in  the  same  colunnn. 


SUBTRACTION. 


173 


//.  Subtract  each  number  in  the  lower  line  from 
the  one  above  it  and  write  the  remainder  in  the 
line  below. 

III.  If  any  number  in  the  lower  line  is  greater 
than  the  one  above  it,  increase  the  latter  by  as 
many  units  as  make  one  of  the  next  higher  de- 
nomination, perform  the  subtraction  and  then 
add  1  unit  to  the  next  number  in  the  lower  line. 


EXAMPLES. 

Perform  the  following  indicated  subtractions : 

(1.)                       (2.)  (3.) 

£     s.  d.  lbs.  oz,   dxvts.  bu.  pks.     qts. 

From  14    14  3      6  11      14  65  1  7 

Take      9     17  1      2  3      16  14  3  4 


Eem.   £4    175. 

2d.    4  lbs.  7  oz.  18  dwts.  50  bu.  2pks.  3  qts. 

Proof. — The  method  of  proof  is  the 
simple  numbers. 

!  same  as  for  subtraction  of 

(4.) 
cwt.    qrs.      lbs. 
7        3         13 

(5.) 
hhds.    gals. 
112        23 

qts. 
1 

(6.) 
yds.     ft.      in. 
4        2        11 

5         1         15 

75        37 

1 

2        2          9 

(7.) 

acres,  sq.  rds. 

29         50 

(8.) 
23°       45'     54" 

(9.) 
fi)    !     3    3   grs. 
35     7    3    1     14 

24        65 

r       49'     57" 

in.  SI 

17  10     6     1     18 

10.  From  4  rds.  2  yds.  1  ft.  9  ^ 

ibtract  2  rds.  3  yds. 

1ft.  11  171. 

174  COMPOUND     LUMBERS. 

OPBBATION. 

rds.  yds,    ft.      in,        rds,     lif.  yds.       in, 
4219     =     4  5  3 

2        3        1      11     =     2  7  5 


1  rd.      8  lif.rds.  16  in, 
z=:  1  rd.      4  yds.  1  ft.  4  in.  Ans. 

11.  From  12  rds.  2  yds,  2  ft.  1  in.  subtract  3  rds.  3  yds. 
2  ft.  10  in. 

12.  From  Srds.  l^yds.  subtract  3  rds.  3  yds.  2  ft.  6  vi. 

Note. — To  write  a  date  as  a  compound  number,  we  first  write 
the  number  of  the  current  year,  then  the  number  of  the  current 
month,  counted  from  the  beginning  of  the  year  (Art.  114),  and 
then  the  number  of  the  day.  Thus,  July  7th,  1839,  is  written  1839 
yrs.  7  mos.  7  da.  In  computing  the  difference  of  two  dates,  a  month 
is  to  be  counted  equal  to  30  days. 

13.  What  is  the  difference  of  time  between  October  16th, 
1869,  and  Aug.  2d,  1873  ? 

OPERATION. 

August  2d,  1873  .  .  .  1873  yrs,    8  mos.    2  da. 
October  16th,  1869 .  .  1869  yrs.  10  mos.  16  da. 

3  yrs.    9  mos.  16  da.    Ans. 

14.  How  long  from  Sept.  25, 1871,  to  July  4,  1876  ? 

15.  How  long  from  July  7, 1815,  to  Nov.  1,  1873  ? 
.16.  How  long  from  May  13,  1816,  to  June  25,  1859? 

17.  What  is  the  difference  between  22  hrs.  17  min.  4  sec, 
and  14  hrs.  9  min.  51  sec.  9 

18.  What  is  the  difference  between  £1.5  and  7s.  6d,f 

19.  From  -^  of  1  hhd.  subtract  f  of  1  qt. 

20.  From  3.107  kilog.  subtract  331.2  grams. 

21.  From  16  da.  21  hrs.  42  min.  13  sec,  subtract  12  da. 
22  hrs,  58  min.  39  sec. 


SUBTRACTION.  176 

32.  From  7  T,  14  cwt  3  qrs.  19  lbs,  subtract  3  T.  18  cwL 
1  gr.  4  lbs, 

23.  From  14  ?5s.  1  oz.  3  <?«/j^5.  18  grs,  subtract  9  lbs,  0  oz, 
16  ^z^^s.  5  ^r5. 

24.  From  4,306  gals.  1  qt,  subtract  3,621  gals.  2  qts,  1  ^^. 

25.  From  110  Z>^.  1  pk.  2  g^s.  subtract  94  bu.  3  ^^5.  7  qts, 

PRACTICAL    PROBLEMS. 

1.  A  merchant  bought  a  piece  of  cloth  for  £22  105.  and 
sold  one  half  of  it  for  £14  18^. ;  for  what  must  he  sell  the 
rest  to  make  £7  14s.  M.  9  Ans.  £15  6s.  Sd. 

2.  A  farm  contains  273  ^.  1  i2.  5  sq.  rds.,  hut  only 
111  ^.  2  i?.  38  sq.  rds.  was  capable  of  tillage;  how  much 
of  it  was  incapable  of  tillage  ? 

3.  From  a  piece  of  cloth  containing  39^  yds.,  there 
was  cut  off  at  one  time  3f  yds.  and  at  another  time  4J- 
yds. ;  how  much  remained  in  the  piece  ? 

4.  A  merchant  has  183  cwt.  24  lbs.  of  butter,  of  which 
he  ships  78  cwt.  3  qrs.  14  lbs. ;  how  much  remains  ? 

5.  How  long  from  Jan.  20,  1873,  to  Nov.  14, 1875  ? 

6.  A  man  was  born  Jan.  10, 1803,  and  died  Sept.  21, 1875 ; 
what  was  his  age  at  the  time  of  his  death  ? 

7.  The  revolutionary  war  began  April  19,  1775,  and 
ended  Jan.  20, 1783  ;  how  long  did  it  last  ? 

8.  How  long  from  the  discovery  of  America,  Oct.  11, 
1492,  to  the  declaration  of  independence,  July  4, 1776  ? 

9.  From  a  pile  of  wood  containing  11  C.  4  C.  ft.,  there 
was  sold  4  (7.  5  C.ft.  12  cu.  ft. ;  how  much  remained? 

10.  The  latitude  of  Albany  is  42°  39'  3"  N.,  and  that  of 
St.  Petersburg  is  59°  56'  N. ;  what  is  the  difference  ? 

Ans.  17°  16'  57". 


176  COMPOUND     NUMBEES. 

Note. — The  Latitude  of  a  place  is  its  angular  distance  from 
the  equator.  If  the  place  is  north  of  the  equator  its  latitude  is 
marked  N.,  if  south,  its  latitude  is  marked  S.  If  the  latitude  of 
two  places  are  both  nortJi  or  both  south,  their  difference  of  latitude  is 
found  bj  subtracting  the  less  from  the  greater  ;  if  the  latitude  of 
one  place  is  north  and  the  other  south,  their  difference  of  latitude  is 
found  by  adding  the  latitudes  of  both. 

11.  The  latitude  of  New  York  is  40°  42'  45"  N.,  that  of 
the  Cape  of  Good  Hope  is  34°  22'  S ;  what  is  the  differ- 
ence? Ans,  75°  4' 45". 

12.  The  latitude  of  St.  Augustin  is  29°  48'  30"  N.,  and 
that  of  Gibraltar  is  36°  7'  N. ;  what  is  the  difference  ? 

13.  The  longitude  of  New  York  is  74^  3'  W.,  and  that 
of  San  Francisco  is  122°  26'  45"  W. ;  what  is  their  differ- 
ence of  longitude  ?  Ans,  48°  23'  45". 

Note. — Longitudes  are  reckoned  both  east  and  west  from  some 
assumed  meridian,  usually  that  of  Greenwich,  England.  The 
method  of  finding  difference  of  longitude  is  the  same  as  for  finding 
difference  of  latitude. 

14.  The  longitude  of  Berlin  is  13°  24'  E.,  and  that  of 
Washington  is  77°  0'  15"  W. ;  what  is  the  difference  ? 

15.  The  longitude  of  Charleston  is  79°  55'  38"  W.,  and 
that  of  Boston  is  71°  3'  30"  W.;  what  is  the  difference  ? 

16.  A  farmer  has  147  hu.  1  pK  of  oats  ;  he  puts  49  lu. 
3  pks.  in  one  bin,  27  lu.  1  pJc.  in  a  second  bin,  32  hu.  3pks. 
in  a  third  bin,  and  the  rest  in  a  fourth  bin ;  how  many 
does  he  put  in  the  fourth  bin  ? 

REVIE^A^    QUESTIONS. 

(151.)  What  is  subtraction  of  compound  numbers?  (152.) 
Give  the  rule.  How  proved  ?  How  are  dates  written  ?  What  is 
the  latitude  of  a  place  ?  How  reckoned  ?  What  is  the  method  of 
finding  difference  of  latitude?    Difference  of  longitude? 


MULTIPLICATION.  177 

V.    MULTIPLICATION    OF    COM- 
POUND   NUMBERS. 

DEFINITION. 

153.  Multiplication  of  Compound  Numbers   is 

the  operation  of  taking  a  compound  number  as  many 
times  as  there  are  units  in  an  abstract  number. 

MENTAL     EXERCISES. 

1.  How  many  inches  are  7  times  8  inches?  What  is 
the  product  of  11  in.  by  9  ?  of  Id.  by  11  ?  of  6  oz.  by  14  ? 
oilQyds.  by  8? 

2.  What  is  the  product  of  M.  by  11  ?  How  many  shil- 
lings in  the  product  and  how  many  pence  remain  ?  What 
is  the  product  of  %ft.  by  16  ?  How  many  yards  and  feet 
in  the  product  ?  What  is  the  product  of  7  qts.  by  13  in 
pecks  and  quarts  9 

3.  What  is  the  product  of  15  in.  by  9  ?    How  m^\ij  feet 

and  inches  in  15  in.9  in  9  times  15  in.9    What  then  is  the 

product  of  1ft.  3  ill.  by  9  ?    What  is  the  product  of  4  iu, 

d2)ks.  by  11?  of  U  7^.  by  8  ? 

Note. — The  principles  used  in  multiplication  of  compound  num- 
bers are  the  same  as  those  used  in  multiplication  of  simple  numbers. 

OPERATION    OF    MULTIPLICATION    OF    COMPOUND    NUMBERS. 

154.  Let  it  be  required  to  multiply  £4  ^s.  6d.  by  16 : 

Explanation. — Having  written  the  operation. 

multiplier  under  the  multiplicand,  we  £         S.        d. 

multiply  5d.  by  16,  which  gives  SOd.,  4          2          5 

or  6s.  8d.  ;  setting  down  8d,  we  carry  ..  ^ 

6s.  to   the    next    column.      We  then  

multiply  2s.  by  16  and  add  6s.  to  the  £65        18s.       Sd. 

product,  which  gives  385.,  or  £1  18s.  ; 

setting  down  18s.,  we  carry  £1  to  the  next  column.    Finallv,  we 


178  COMPOUND     NUMBERS. 

multiply  £4  by  16  and  add  £1  to  the  product,  which  gives  £65. 
Hence,  the  required  product  is  £65  18s.  Sd. 

In  like  manner  we  may  treat  all  similar  cases ;  hence,  the  fol- 
lowing 

RULE. 

/.  Multiply  the  units  of  the  lowest  denomina- 
tion of  the  multiplicand  by  the  multiplier,  and 
divide  the  product  by  the  number  of  the  scale 
that  connects  this  denomination  with  the  next 
higher  one;  set  down  the  remainder  and  carry 
the  quotient  to  the  next  column. 

II.  Multiply  the  units  of  the  next  higher  de- 
nomination by  the  multiplier,  add  the  units 
brought  forward,  and  proceed  as  before,  continu- 
ing the  operation  till  all  the  parts  of  the  given 
number  have  been  multiplied, 

EXAMPLES. 

(1-)  (3.) 

£          8,  d,  cwt      qrs,       lbs.      oz. 

Multiplicand.     17         15  9  8           3           19 

Multiplier. . .  6  7 

Product ....  £106        145.      ed,       61  cwt,  1  qr,  10  lbs.  15  oz, 

(3.) 
mi.    rds.   yds.  ft.    in.     mi.        rds.     Tif.  yds.       in, 
9      110      4      2      6  =  9  110  9  12 

9 9 

84  mi.      37  rds.  10  hf.  yds.  0  in. 
=  84  mi.  37  rds.  5  yds.  Ans, 

Multiply  Multiply 

4.  5  mot.  2  qrs,  by  7.  5.  $8.75  by  24.5. 


MULTIPLICATION.  179 

6.  65.35//-.  by  46.  14.  £|  by  17.5. 

7.  5Mkilog.  by  12.  15.  10^.  1  M.  by  11. 

8.  15  yds.  1ft,  by  21.  16.  3  lirs,  15\min.  by  24. 

9.  123.25  m.  by  15.  17.  5  ciot.  2|  qrs.  by  24. 

10.  6  whs,  3  da.  by  13.  18.  £3  14f<?.  by  33. 

11.  11  gals.  2  qts.  by  18.  19.  18  T.  5  cwt  by  127. 

12.  15  lbs.  3  02!.  by  16.  20.  A:  yds.  ^ft.  by  18.75. 

13.  13  yds.  2ift.  by  9.  21.  11  gals.  3  ^2^5.  by  14.72. 

22.  How  many  square  feet  in  a  rectangle  whose  length 
is  17/^.  5  m.j  and  whose  breadth  is  Sft.  9in.9 

Ans.  17.417  x  3.75  =  66.dl4:sq.ft. 

Note. — In  examples  like  the  above,  we  reduce  both  dimensions 
to  decimals  of  a  foot,  as  explained  in  Art.  148  ;  we  then  multiply 
the  number  of  feet  in  the  length  by  the  number  of  feet  in  the 
breadth  ;  the  product  will  be  the  number  of  square  feet  in  the  area 
(Art.  122). 

To  secure  uniformity  in  this  and  all  similar  cases,  let  the  student 
carry  decimals  to  three  places,  applying  the  rule  for  approximation, 
given  in  Art.  90,  at  each  step  of  the  operation. 

23.  How  many  square  yards  in  the  floor  of  a  room 

whose  length  is  22//.  4^?^.,  and  whose  breadth  is  16//. 

din.f 

Solution.— Because  22  ft.  4w.  =  7.444 y<fe.,  and  16//.  9  m.  = 
5.583  yds.,  we  have  7.444  x  5.583  =  41.561  sq.  yds.    Ans. 

24.  How  many  sq.  ft.  in  a  floor  16//.  6  in.  long  and 
13//.  9 171.  wide  ? 

Solution.— 16.5/i{.  x  13.75/^  =  22Qi  sq.  ft.  Ans. 

25.  How  many  sq.  yds.  in  a  plot  102  yds.  long  and 
Uyds.  2ft.  wide? 

26.  How  many  sq.  rds.  in  a  field  4:2^  rds.  long  and 
Urds.  wide? 


180  COMPOUND     NUMBERS. 

27.  How  many  cubic  feet  in  a  bin  1ft.  2  in.  long,  ^ft. 
4  in.  wide,  and  %ft.  9  in.  deep  ? 

Ans.  7.167  x  3.333  x  2.75  =  65.682  c^. /if. 

Explanation. — Here  we  reduce  each  dimension  to  the  required 
linear  unit,  and  then  find  the  continued  product  of  the  correspond- 
ing numbers.    First,  we  find  7.167  x  3.333  =  23.888  ;  we  then  find 

23.888x2.75  =  65.682, 

28.  How  many  cuUc  yards  in  rectangular  volume  of 
earth  27  yds.  1ft.  long,  13  yds.  2ft.  wide,  and  2  yds.  high  ? 

29.  How  many  cubic  feet  in  a  hewn  stick  of  timber  27  ft. 
long,  1ft.  2  in.  wide,  and  9  in.  thick  ? 

30.  How  many  cuMc  feet  in  a  room  12^//.  long,  10^ ft. 
wide,  and  lift,  high  ?  How  many  square  feet  in  the 
floor  ? 

31.  What  is  the  product  of  lift.  9  in.  by  14//.  6  in. 

PRACTICAL    PROBLEMS. 

1.  If  a  gentleman  spends  <£1  Is.  6d,  a  day,  how  much 
will  he  spend  in  365  days  ? 

2.  What  is  the  length  of  36  pieces  of  telegraph  wire,  the 
length  of  each  piece  being  26  mi.  1,125  yds.  9 

3.  What  is  the  weight  of  37  parcels  of  silver,  averaging 
12  lis.  2  02.  15  dwts.  6  grs.  each  ? 

4.  How  many  yards  of  cloth  in  27  bales,  each  bale  con- 
taining 15  pieces,  and  each  piece  15|  yds.  9 

5.  In  7  loads  of  wood,  each  containing  2  C  3  C.ft.,  how 
many  cords  ?    Kow -majiy  cubic  feet  9 

6.  What  is  the  weight  of  1|  dozeji  silver  forks,  each  fork 
weighing  2  oz.  3  dtvts.  9 

7.  What  is  the  weight  of  7  hJids.  of  sugar,  each  hogshead 
weighing  5  cwt.  1  qr.  14  lbs.  9 


MULTIPLICATION^.  181 

8.  How  far  can  a  man  travel  in  5  days,  at  the  rate  of 
7^  mL  a  day  ? 

9.  How  much  land  is  there  in  9  fields,  each  containing 
^A.2R.  30  sq.  rds.  9 

10.  What  do  54  sheep  cost  at  15s.  3d  each  ? 

11.  How  many  yds.  in  12  pieces,  each  containing  19f 
yds  J 

12.  What  is  the  cost  of  52|  yds,  of  cloth  at  $3.50  a  yard? 

13.  How  many  yds.  in  GJ  pieces  of  muslin,  each  contain- 
ing 39J  yds.  ? 

14.  What  is  the  weight  of  15  loads  of  hay,  each  weigh- 
ing 1  7^.  3  civts.? 

15.  How  many  sq.  yds.  of  flagging  will  be  required  to 
flag  a  court  1V1  ft.  long  and  98/^.  3  i7i.  wide  ? 

16.  How  many  square  feet  of  boards  will  be  required  to 
make  a  floor  22//.  6  in.  long  and  19//.  9  in.  broad? 

17.  A  room  is  18^//.  long,  13^//.  wide,  and  9^//.  high  ; 
how  many  square  yards  of  carpetin^  will  it  require  to 
cover  the  floor  ?  How  many  square  feei  of  kalsomining 
in  the  ceiling  ?  How  many  square  yards  of  paper  will  it 
take  to  cover  the  walls,  the  doors  and  windows  not  being 
taken  into  account  ? 

18.  A  rectangular  block  of  stone  is  Q  ft.  long  3//.  wide, 
and  24  ft.  thick ;  what  is  its  weight,  if  each  cubic  foot 
weighs  156  lbs.  ? 

REVIEW    QUESTIONS. 

(153.)  What  is  multiplication  of  compound  numbers  ?  (154.) 
Give  rule  for  multiplication  of  compound  numbers  ?  How  do 
you  find  the  contents  of  a  rectangular  area?  Of  a  rectangular 
solid? 


182  COMPOUND     NUMBEB8. 

VI.    DIVISION    OF    COMPOUND 
NUMBERS. 

DEFINITION. 

155.  Division  of  Compound  Numbers  is  the  oper- 
ation of  dividing  a  compound  number  by  an  abstract  num- 
ber, or  by  a  similar  denominate  number. 

MENTAL      EXERCISES. 

1.  If  35  nuts  are  divided  equally  among  7  boys,  how 
many  will  each  receive  ?  AVhat  is  the  quotient  of  35  nuts 
by  7  ?  Of  IM.  by  6  ?  Of  27  horses  by  9  ?  Of  48  yds.  by 
6?     Of  156  52^.  by  13  ?     Of  156  by  13  ? 

2.  If  27  marines  are  divided  in  piles,  each  containing  9 
marUes,  how  many  piles  will  there  be  ?  What  is  the  quo- 
tient of  27  marbles  by  9  marbles  f  Of  49  rods  by  7  rods  9 
Of  965.  by  125.  ?    Of  180^.  by  15^.  9    Of  180  by  15  ? 

3.  What  is  the  quotient  of  153^?.  by  9  ?  How  many 
shillings  in  1536?.,  and  how  many  pence  remain  ?  How 
many  shillings  and  pence  in  17d.  9  What  then  is  the  quo- 
tient of  12s.  9d.  by  9  ?  What  is  the  unit  of  153^7.  9  Of 
9d.?    Ofl7d.9 

4.  What  is  the  quotient  of  84  yds.  by  12  yds.  9  What  is 
the  unit  of  84  yds.  9  Of  12  yds.  9  Of  7  ?  Is  tliere  any 
difference  between  the  quotient  of  57  lbs.  by  19  lbs.  and  of 
57  by  19  ? 

5.  A  floor  contains  40  square  yards,  and  its  length  is 
5 yards;  what  is  its  breadth?  What  is  the  quotient  of 
4  square  yards  by  5  lineal  yards  ? 

Note. — The  principles  used  in  division  of  compound  numbers 
are  the  same  as  those  used  in  division  of  simple  numbers. 


OPBEATION. 

£        s. 

d. 

16 

)6o      18 

8  (  £4  2s. 

5d. 

64 

£1  .  .  . 

1st  rem. 

20 

38s. 

32 
6s.  .  . 

.  2dLrem. 

12 

80^. 

80 

DIVISION.  183 

OPERATION    OF    DIVISION    OF    COMPOUND    NUMBERS. 

156.  Let  it  be  required  to  divide  £65  18s.  Sd.  by  16  : 

Explanation.  —Dividing 
£65  by  16,  we  find  a  quo- 
tient £4,  and  a  remainder 
£1.  Reducing  £1  to  shil- 
lings and  adding  18s.,  we 
have  38s.,  which  we  take 
for  a  new  dividend  ;  divid- 
ing 38s.  by  16,  we  find  a 
quotient  2s.  and  a  remain- 
der Qs.  Reducing  6s.  to 
pence  and  adding  Sd.,  we 
have  80d.,  which  we  take 
for  a  new  dividend  ;  divid- 
ing 80d.  by  16,  we  find  the 
quotient  5d.  and  a  remain- 
der 0.  The  required  quo- 
tient is  therefore  £4  2s.  5d. 

In  like  manner  we  may  treat  all  similar  cases  ;  hence,  the  fol- 
lowing 

RULE. 

/.  Divide  the  units  of  the  highest  denomination 
in  the  dividend  by  the  divisor  and  write  the  quo- 
tient as  a  part  of  the  required  quotient. 

II,  Reduce  the  remainder  to  the  next  lower 
denomination,  and  to  the  result  add  the  units  of 
that  denomination,  for  a  new  dividend,  and  pro- 
ceed as  before. 

III.  Continue  this  operation  till  the  division  is 
completed. 

EXAM  PLES. 

Perform  the  following  indicated  divisions : 
(1.)  (2.) 

7 )  37  lu.  3  pTcs.  7  qts.        9  )  1  T.  19  cwt  2  qrs.  12  lbs. 


Quotient,  5  hu.  1  pk.  S^  qts.  4  cwt.  1  qr.   15f  Tbs, 


184  COMPOUND    KUMBERS. 

3.  Divide  17  cwt.  0  qrs.  2  lbs.  C  oz.  by  7. 

4.  Divide  228  T.  18  cwL  3  qrs.  13  /^5.  12  oz.  by  11. 

5.  Divide  9  hhds.  28  ^«Zs.  2  (//s.  by  49  ^«/s.  2  g'^fs.  1  ;?^. 

Explanation. — The  dividend  is  equal  to  4764  pts.  and  tlie  divi- 
sor is  equal  to  397  pts.  ;  hence,  the  quotient  is  equal  to  4764  pts.  -r- 
397  pts.  =  12.  Ans.    All  similar  cases  may  be  solved  by  the  following 

RULE. 

Reduce  both  numbers  to  the  same  denomina- 
tion and  divide  as  in  simple  numbers. 

Note. — If  the  divisor  is  abstract,  the  quotient  is  similar  to  the 
dividend  ;  if  the  divisor  is  similar  to  the  dividend,  the  quotient  is 
abstract. 

6.  Divide  17  lea,  1  mi.  4:  fur.  21  rds.  by  21. 

7.  Divide  25  bu,  3  j^ks.  4  qts.  by  9. 
Perform  the  following  indicated  divisions : 

8.  £21  lis,  M.  by  15.  20.  129^  bu.  by  8  hi.  3  qts, 

9.  15  Ws.  3  oz.  12  dwts,-^12.  21.  58650  m.  -^  3.45  Mom. 

10.  39  kilog.  by  7.5.  22.  56°  45'-^  15°. 

11.  3  mi.  by  22  ft.  6  in.  23.  78°  16'  15"-^  15. 

12.  1361  mi,  188  rds,  -^  28.  24.  69°  4'  45"-r- 15. 

13.  117.9 /r.  by  131.  25.  2  gals.  3  qts.^  3  qts, 

14.  203  hectol  by  58.  26.  3  T.  16  cwt.-^  19  lbs. 

15.  £30.  7s.  li^.  by  7.  27.  12  hrs,  48  m^7^.-^  16  sec, 

16.  9  /J?^6'.  2Sigals.  by  12.  28.  15.75  «/^5.-t-  3.5  yds. 

17.  3  m^.  by  2  ft.  3  m.  29.  24  rds.  3  i/^s.-j-  27//. 

18.  186.02 /r.  by  131.  30.  48°  4'  30" -4-  45°. 

19.  39.06  Jcilog.  by  93.  31.  £6  Is.  6d.-^27. 

Note. — When  the  divisor  is  composite,  we  may  divide  by  each  of 
its  factors  in  succession.  In  the  last  example,  if  v^^e  divide  by  3, 
we  find  for  a  quotient  £2  Os.  Qd. ;  and  this  result  divided  by  9 
gives  As.  6d.    Ans. 


DIVISION.  185 

32.  Divide  £37  14s.  by  24,  that  is,  by  4  x  6. 

33.  Divide  178  Ihs,  9  oz,  14  divh.  16  gr.  by  77. 

34.  Divide  147  hiu  3  pks.  4  ^/s.  by  5  hu. 

35.  Divide  2  T.  5  cwt.  24  /^s.  by-12  ^^5. 

PRACTICAL    PROBLEMS  .—Miscellaneous. 

1.  If  7  calves  cost  £15  1.?.,  what  is  the  cost  of  1  calf  ? 
of  5  calves  ?  of  13  calves  ? 

2.  If  a  man  can  walk  38 J  miles  in  11  hours,  how  far  can 
he  walk  in  3  hours  ?  in  9  hours  ? 

3.  How  many  times  can  a  3-quart  measure  be  filled  from 
a  cask  of  wine  containing  26  gals.  1  qt  ? 

4.  How  many  yds.  of  cambric  f  yd.  wide  will  it  take  to 
line  14  yds.  of  cloth  1^  yds.  wide  ? 

5.  A  garden  containing  1,154J  sq.  yds.  is  40^  yds.  long ; 
how  wide  is  it  ? 

6.  A  stick  of  hewn  timber  containing  30  cu.ft.  is  32/i^. 
long  and  1ft.  3  in.  wide;  what  is  its  thickness? 

7.  What  is  the  cost  of  1^  mi.  of  iron  pipe  at  12^  cts. 
a  foot  ? 

8.  How  many  panes  of  glass,  each  12  in.  by  15  in.,  in  a 
box  of  glass  containing  100  sq.ft.  9 

9.  If  4  qts.  of  salt  cost  21  cts.,  what  is  the  cost  of 
4  bu.  3  pks.  9 

10.  What  will  it  cost  to  carpet  a  floor  IS  ft.  long  and 
15|//.  wide,  with  carpeting  iyd.  wide  and  costing  $lf 
a  yard  ? 

11.  A  room  is  24:  ft.  long,  l^ft-  wide,  and  10|//.  high ; 
what  will  it  cost  to  plaster  the  4  sides  of  the  room  at  the 
rate  of  30  cts.  a  sq.  yd.  after  deducting  |  for  doors  and 
windows  ? 


186  COMPOUND     NUMBERS. 

12.  What  will  it  cost  to  plaster  and  kalsomine  the  ceil- 
ing of  the  room  described  in  Problem  11,  at  40  cts.  a 
square  yard  ? 

13.  What  will  it  cost  to  carpet  the  same  room  with  car- 
peting 1  yd.  wide  at  $2.50  a  yard  ? 

14.  A  walk  ^ft.  wide  and  180/^.  long  is  to  be  flagged 
with  stones  18  in.  long  and  9  in.  wide ;  how  many  will  it 
take  ? 

15.  How  many  rolls  of  paper,  each  19/^.  long  and  1ft, 
6 171.  wide,  will  it  take  to  paper  the  sides  of  a  room  18/if. 
long,  15/if.  wide,  and  9J/if.  high,  no  allowance  being  made 
for  doors  or  windows  ? 

16.  A  man  divides  43^.  ^  R.  20sq.rds.mto  18  equal 
building  lots ;  how  much  in  each  lot  ? 

17.  A  farm  of  154^.  20  5g.  rds.  is  laid  out  in  lots  con- 
taining 12  ^.  Z  R.  15  sq.  rds.  each ;  how  many  lots  are 
there  ? 

18.  A  merchant  bays  10  cwt  2  qrs.  of  sugar  for  $131.25, 
for  what  must  he  sell  it  to  make  1^  cts.  a  pound  ? 

19.  A  speculator  bought  4  city  lots,  each  25  by  100  feet, 
at  %1^  a  square  foot,  and  sold  them  again  for  120,470; 
what  did  he  gain  on  each  lot  ? 

20.  If  20  bricks  will  build  1  cubic  foot  of  wall,  how 
many  bricks  will  be  required  to  build  a  wall  70//.  long, 
l^ft.  thick,  and  ^ft.  high? 

21.  How  many  sheets  of  tin,  18  in,  long  and  15  in.  wide, 
will  it  take  to  cover  the  roof  of  a  house  40  feet  long,  the 
rafters  on  each  side  being  18  feet  long  ? 

22.  What  will  it  cost  to  floor  a  room  Viift.  long  and 
16 /if.  wide,  at  the  rate  of  $1.10  per  square  yard  ? 


DIVISION.  187 

23.  If  it  costs  $62.50  to  lay  50  sq.  yds.  of  flooring,  what 
will  it  cost  to  lay  a  floor  33/i^.  long  and  18/if.  wide  ? 

24.  How  many  spoons,  each  weighing  %oz.  10  dwfs.,  can 
be  made  from  a  bar  of  silver  weighing  11  lis.  3  oz.  9 

25.  The  circumference  of  the  fore  wheel  of  a  carriage  is 
13/i^.  dill,  and  that  of  the  hind  wheel  IQft.  Qm.;  how 
many  times  more  will  the  fore  wheel  turn  than  the  hind 
one  in  a  jonrney  of  30  miles  ? 

26.  If  160  bushels  of  oysters  cost  £75  17^.  4^.,  what 
does  1  bushel  cost  ? 

27.  A  truckman  carried  117  G.  110  cu.  ft.  of  wood  in  100 
equal  loads ;  how  much  did  he  carry  at  each  load  ? 

28.  If  3  yds.  of  cloth  cost  £4  16^.  %d.,  how  much  does 
1  yard  cost,  and  how  much  does  12  yds.  cost  ? 

29.  A  person's  yearly  income  is  14,636.80/r. ;  of  this  he 
gives  in  charity  2,500/r.;  his  weekly  bills  are  149.15/r. 
each ;  and  the  rest  he  spends  in  traveling ;  how  much  does 
he  spend  per  week  in  traveling  ? 

30.  The  average  speed  of  a  railway  train  is  ^myriameters 
per  hour ;  how  long  will  it  take  to  travel  from  Paris  to 
Boulogne,  a  distance  of  272  kilom.  ? 

31.  Bought  the  following  articles :  27|-  meters  of  linen 
at  3.50/r.  per  meter;  Q\ meters  of  velvet  at  17/r.  per 
meter;  2*^  meters  of  brocade  at  16.75 /r.  per  meter; 
28f  meters  of  merino  at  4/y\  per  meter ;  1^  doz. pairs  of 
socks  at  3.05 /r.  per  pair;  and  "1  pairs  of  gloves  at  42 /r.  a 
dozen  pairs ;  what  was  the  amount  of  the  bill  ? 

32.  If  a  person  takes  108  steps  a  minute,  each  step  being 
30  inches,  how  far  can  he  walk  in  2  hrs.  30  min.  9  How 
far  can  he  walk  in  3  hrs.  10  min.  9 


188  COMPOUND     NUMBERS. 

33.  How  many  yards  of  cloth  will  it  take  to  clothe  a 
company  of  48  men,  if  39|  yds.  will  clothe  7  men  ? 

34.  The  longitude  of  Philadelphia  is  75°  9'  23"  W.; 
that  of  San  Francisco  is  122°  24'  39"  W.:  what  is  the 
difference  ? 

35.  The  latitude  of  Duhlin  is  53°  23'  N.;  that  of  Santi- 
ago is  33°  26'  26"  S.:  what  is  the  difference  ? 

APPLICATION    TO    LONGITUDE    AND    TIME. 

157.  From  the  explanation  of  longitude  in  Art.  137, 
it  appears  that  the  sun  comes  to  the  meridian  of  a  place 
whose  longitude  is  15°  W.,  1  hour  later  than  it  comes  to 
the  standard  meridian.  Hence,  the  local  times  at  the  two 
meridians  differ  by  1  hour ;  at  places  whose  longitudes 
differ  by  30°  the  difference  of  local  times  is  2  hours  ;  and 
so  on,  according  to  the  Table  in  Art.  137. 

EXAM  PLES. 

1.  Let  it  be  required  to  find  the  difference  of  time  at 
two  places  where  difference  of  longitude  is  74°  1'. 

Explanation.  —  Dividing  74                      opbration. 
by  15,  we  find  a  quotient  4,  and          -^v^  \  ^4°  '\^ 
a  remainder   14°  :    we   call  the  


quotient  \lirs.  :   reducing  14°  to  4:^r5.  h^min.  ^sec, 

minutes  of  arc  and  adding  1',  we 

have  841',  which  divided  by  15  gives  a  quotient  56,  and  a  remain- 
der V  ;  we  call  the  quotient  minutes  of  time,  56  min.;  reducing  1'  to 
seconds  of  a/rc,  we  have  60",  which  divided  by  15  gives  4;  this  we 
call  seconds  of  time,  4  sec.  Hence,  the  required  difference  of  time  is 
4  hrs.  56  min.  4  sec. 

By  reversing  the  process  just  explained,  we  can  find  the  difference 
of  longitude  of  two  places  when  we  know  the  difference  of  their 
local  times. 

Hence,  the  following 


DIVISION".  189 

RULE. 

1°.  Divide  the  difference  of  longitude  in  arc  by 
15,  changing  degrees,  ininwtes,  and  seconds  of  arc 
to  hours,  minutes,  and  seconds  of  time ;  the  result 
luill  he  the  difference  of  time. 

2°.  Multiply  the  difference  of  time  by  Id, 
changing  hours,  jninutes,  and  seconds  of  time  to 
degrees,  minutes,  and  seconds  of  arc;  the  result 
will  be  the  difference  of  longitude  in  arc. 

Keduce  Reduce 

2.  17°  24'  15"  to  time.  8.  3  hrs,  4:min.  6 sec,  to  arc. 

3.  54°  18'  45"     "  9.  2  hrs.  9min.  IS  sec,  " 

4.  118°  23'  30"  "  10.  6  hrs.  Umin.  2d  sec.  " 

5.  21°  47'  45"    "  11.  9 hrs.  Umin.  10 sec.  " 

6.  79°  40'  15"     "  12.  Ihr.  IS min.  dQsec.  " 

7.  38°  38'  45"     "  13.  6  hrs.  29  mm.  4  sec.  " 

14.  The  difference  of  longitude  between  Rochester,  N.Y., 
and  San  Diego,  Cal.,  is  39°  22'  22" ;  what  is  their  difference 
of  time  ? 

15.  The  difference  of  time  between  Liverpool  and  West 
Point  is  4:  hrs.  43  min.  4:Ssec.;  what  is  their  difference  of 
longitude  ? 

REVIE^A^     QUESTIONS. 

(155.)  Wliat  is  division  of  compound  numbers?  (156.)  Give 
the  rule  when  the  dividend  is  denominate  and  the  divisor  abstract. 
When  the  dividend  and  divisor  are  similar  denominate  numbers. 
(157.)  How  do  you  reduce  difference  of  longitude  to  difference  of 
time  ?    Diflfereuce  of  time  to  difference  of  longitude  ? 


/ 


I.  PERCENTAGE. 

DEFINITIONS. 

158.   Per  Cent,  means  by  the  hundred, 

or  Hundredths.    Thus,  7 per  cent,  of  $100 

is  H  hundredths  of  1100,  or  I7. 

The  Sign,  %,  is  read  per  cent.    Thus,  7%  of  $100  is  read  7  per 
cent,  of  $100. 

159.  The  Rate  Per  Cent.,  or  simply  the  rate,  is  the 

number  of  hundredths  taken.     Thus,  in  the  expression 

7%  of  1100,  the  rate  is  7  hundredths,  or  .07. 

Note. — Bate  per  cent,  may  be  expressed  in  any  of  the  ways  chowp 
in  the  following 

TABLE     OF     EQUIVALENTS. 

7  per  cent.,  or     7  ^,  is  equivalent  to  y^^,  or  to    .07. 
\2\  per  cent.,  ox   12\%,   "  "     ^^,    "      .125. 

100  per  cent.,  or  1{)^  %,   "■  **     \^%,      "    1. 

125    per  cent.,  or  12^  %,    "  *'     {^,      "    1.25. 


etc.,  etc.,  etc.,  etc. 

160.  Percentage  is  some  per  cent,  of  a  given  number. 

Thus,  $7  is  the  percentage  on  $100  when  the  rate  is  7  per 

cent. 

Note. — The  general  term  percentage  is  applied  to  all  operations 
in  which  the  computation  is  made  by  hundredths. 


PERCENTAGE.  191 

161.  The  Base  is  the  number  on  which  percentage  is 
reckoned.  Thus,  in  the  expression  ^%  of  $100,  the  lase  is 
$100. 

MENTAL      EXERCISES. 

1.  How  many  per  cent,  is  4z  hundredths  9  .09?  .74? 
.12 J?  How  many  hundredths  is  11^?  15^?  37^^? 
150^? 

2.  What  is  4  hundredths  of  $100  ?  What  is  4^  of  $100  ? 
b%  of  180  ?     9^  of  16  lbs.?    12^  of  l^ft.  ?    11^  of  $14  ? 

3.  What  is  1%%  of  20«/«fs.f  What  is  the  iase?  The 
rate  per  cent  J  The  percentage?  What  is  37^^  of  $100  ? 
What  is  the  base  ?  The  rate  ?  The  percentage  ?  If  you 
multiply  the  base  by  the  rate,  what  is  the  product  ? 

ADDITIONAL     DEFINITIONS. 

163.  The  Amount  is  the  base  increased  by  the  per- 
centage. Thus,  the  amount  of  $100  increased  by  8^,  is 
$100  +  $8,  or  1108. 

163.  The  Difference  is  the  base  diminished  by  the 

percentage.     Thus,  the  difference  of  $100  diminished  by 

8^  is  $100  —  $8,  or  $92. 

Note. — Both  amount  and  difference  are  'percentages.  Thus,  the 
amount  of  $100  increased  by  8%  is  108%  of  $100  ;  and  the  difference 
of  $100  diminished  by  8%  is  92%  of  $100. 

MENTAL      EXERCISES. 

1.  What  is  h%  of  40  lbs.  9  If  40  lbs.  is  increased  by  b% 
of  40  lbs.,  what  is  the  amount  ?  What  per  cent,  of  40  lbs. 
is  the  amount  ?  What  is  the  amount  of  $80  increased  by 
\b%  of  $80  ?    What  per  cent  of  $80  is  $92  ? 

2.  What  is  7^  of  $60?  If  $G0  is  diminished  by  7^  of 
$60,  what  is  the  difference  ?    How  many  per  cent,  of  $60 


192  PERCENTAGE     AI^D     ITS     APPLICATIOKS. 

is  the  difference  ?    What  is  the  difference  of  30  yds.  and 
20^  of  302/^5.  f    What  per  cent,  of  30  ^/^5.  is  24?/c?5.  F 

3.  The  base  is  120  and  the  rate  is  7^,  what  is  the 
amount  ?  The  base  is  15  lbs.  and  the  rate  Q%,  wliat  is  the 
difference  ?  The  base  is  20  yds.  and  the  rate  10^,  what  is 
the  amount  and  what  is  the  difference  ? 

4.  A  man  had  30  chickens,  but  20^  of  them  were  de- 
stroyed by  a  fox ;  how  many  per  cent,  were  left  ?  How 
many  chickens  were  destroyed  ?  How  many  chickens 
were  left  ?    What  is  m%  of  30  chickens  ? 

PRI  NCI  PLES. 

164,  From  what  precedes  we  have  the  following  prin- 
ciples : 

1°.  The  percentage  is  equal  to  the  base  multiplied  by  the 
rate. 

2°.  The  amount  is  equal  to  the  base  multiplied  by  1  plus 
the  rate. 

3°.  TJie  difference  is  equal  to  the  base  multiplied  by  1 

minus  the  rate. 

Because  either  of  two  factors  is  equal  to  their  product  divided  by 
the  other,  we  have  the  following  principles  : 

4°.  The  rate  is  equal  to  the  percentage  divided  by  the 
base. 

6°.  The  base  is  equal  to  the  percentage  divided  by  the 
rate;  to  the  amount  divided  by  1  plus  the  rate;  or,  to  the 
difference  divided  by  1  minus  the  rate. 

Note. — The  following  rules  are  deduced  immediately  from  the 
foregoing  principles. 

165.  To  find  the  Percentage  when  the  Base 
and  Rate  are  given. 


PERCENTAGE.  193 

RULE. 

Multiply  the  base  by  the  rate, 

EXAMPLES. 

1.  What  is  25^  of  40  lbs.  9  Ans,  40  lbs,  x  .25  =  10  lbs. 
What  is 

2.  9%  of  711  lbs.  9  10.  d.Hfo  of  140.5  ? 

3.  Si%  of  11,200  ?  11.  125^  of  $14.40  ? 

4.  7^  of  810  yds.?  12.  210^  of  97.4  kUo7n.f 

5.  3|^  of  392  miles?  13.  93^^  of  4.56  lbs.  f 

6.  6^  of  1500  ?  14.  624^  of  $185. 57  ? 

7.  12^  of  $^  ?  15.  15|^  of  1136.64  ? 

8.  11^  of  31.25  Mog,  9  16.  25%  of  £.37  ? 

9.  14%  of  875  ft.  9  17.  18|%  of  84  yds.  9 

18.  A  man's  income  is  11700  a  year,  and  he  spends  75^ 
of  it;  how  many  dollars  does  he  spend?       Ans.  11275. 

19.  A  bought  320  acres  of  land,  and  sold  62^%  of  it; 
how  many  acres  did  he  sell  ? 

20.  What  is  75%  of  £28  I65.  M.  ? 

166.  To  find  the  Amount,  or  the  Difference, 
when  the  Base  and  the  Rate  are  given. 

RULE. 

To  find  the  amount,  multiply  the  base  by  1  plus 

the  rate.    To  find  the  difference,  multiply  the  base 

by  1  minus  the  rate. 

Note. — The  amount  can  be  formed  by  adding  the  percentage  to 
the  base,  and  the  difference  by  subtracting  the  percentage  from  the 
base. 

EXAM  PLE  S. 

1.  What  is  the  amount  of  694  Tbs.  increased  by  10^  of 
694  Tbs.  9  Ans,  694  lbs.  x  1.10  =  763.4  lbs. 


194  PERCENTAGE    AND     ITS    APPLICATIONS. 

2.  A  paid  1175  for  a  horse,  and  sold  him  at  an  advance 
of  20^ ;  what  was  the  selling  price  ? 

Ans.  1175+20^  of  $175  =  $210. 

3.  If  464  yds,  is  diminished  by  16%  of  itself,  what  is  the 
difference  ?  Ans,  464  yds,  x  .84  =  389.76  yds, 

4.  A  farmer  had  72  tons  of  hay,  and  sold  16f^  of  it; 
how  much  had  he  left  ? 

Ans.  72  ^.— 16|^  of  72  T.=  60  T. 
What  is  the  amount  and  what  is  the  difference  of_ 

5.  $200  and  9%  of  $200  ? 

6.  770  lbs,  and  25^  of  770  lbs,? 

7.  48  yds.  and  33^^  of  48  yds,  9 

8.  42  Mlom.  and  8%  of  42  Mlom,f 

9.  75  hhds.  and  66f%^  of  75  hMs,9 

10.  36  yrs.  and  12^  of  36  yrs,  9 

11.  152 /if.  and  7.5^  of  U%fU 

12.  67  mi.  and  11^  of  67  mi,9 

13.  64  r.  and35i^of64r.f 

14.  $86  and  16^  of  $86  ? 

15.  72  lu.  and  9%  oi  72  bu.  9   16.  56/r.  and  18^  of  56/n 

17.  A  man  bought  a  watch  for  $115,  and  sold  it  at  a 
loss  of  20^ ;  what  was  the  selling  price  ? 

18.  Of  a  farm  containing  118  A.,  55^  is  arable,  and  the 
rest  is  woodland ;  how  many  acres  of  woodland  ? 

19.  If  cloth  cost  $4.50  a  yard,  for  how  much  must  it  be 
sold  per  yard  to  gain  25^  ? 

20.  A  grocer  bought  sugar  at  10  ds.  a  pound,  and  sold 
it  at  a  loss  of  15^;  what  was  the  selling  price  per  pouod  ? 

167.  To  find  the   Rate  when  Base  and  Per- 
centage are  given. 


PERCENTAGE.  '  196 

RULE. 

Divide  the  percentage  by  the  base. 

EXAMPLES. 

1.  The  percentage  is  $90.24,  and  the  base  is  $752 ;  what 

is  the  rate?  Ans.  |90.24-^$752=. 12=12^. 

2.  A  merchant  bought  a  sloop  for  16,250,  and  sold  it 
again  for  $7,750;  what  did  he  gain  per  cent?  Ans  24^. 

Explanation.— Here  the  percentage  is  $7750  —  $6250  =  $1500, 
and  the  base  is  $6250  ;  hence,  the  percentage  is  $1500  ^  $6250  =.24. 

3.  A  merchant  bought  cloth  at  $5  a  yard,  and  sold  it 

at  $4.50  a  yard;  how  much  did  he  lose  per  cent.  ? 

Ans.  1Q%. 
What  per  cent  of 

4.  $50  is  $3  ?  8.  4  mi.  is  140  rds.  9 

5.  £16  is  £7  ?  9.  8.25 /r.  is  37.96 /r.  ? 

6.  100  yds  is  1|  yds.?  10.  407i  iu.  is  505.3  hii.f 

7.  £1  is  2s.  M.  9  11.  1,248  yds.  is  2,080  yds.? 

12.  A  grocer  bought  1,140  lbs.  of  sugar  at  'd^cts.  a  pound, 
and  sold  the  whole  for  $129.96  ;  what  did  he  gain  per  cent.  ? 

13.  A  farmer  raised  250  hu.  of  oats,  and  sold  all  but 
75  iu. ;  what  per  cent,  of  his  crop  did  he  sell  ? 

14.  A.  bought  216  yds.  of  muslin  at  llfcts.  a  yard,  and 
sold  the  lot  for  $31.72J ;  how  many  per  cent,  did  he  gain  ? 

15.  The  property  of  a  bankrupt  is  worth  $6,102.95,  and 
his  debts  amount  to  $8,225 ;  what  per  cent,  can  he  pay  ? 

16.  In  a  journey  of  1,664  mi.,  A.  travels  208  mi.  by  stage 
and  the  rest  by  rail ;  what  per  cent,  does  he  travel  by  rail  ? 

17.  A  man's  salary  is  $4,200;  of  this  he  spends  22^  for 
fuel  and  rent,  15^  for  clothing,  and  $1,218  for  other  pur- 
poses :  what  per  cent,  of  his  salary  has  he  left  ? 


196  PERCENTAGE     AND     ITS     APPLICATIONS. 

18.  Out  of  a  cask  containing  QQ^gals.  26,6  gals,  were 
drawn  ;  what  per  cent,  remained  in  the  cask  ? 

168.  To  find  the  Base  when  the  Rate  and 
Percentage  are  given. 

RULE. 
Divide  the  percentage  hy  the  rate. 

Note. — If  the  rate  and  amount  are  given,  divide  the  amount  by 
1  plus  the  rate  ;  if  the  rate  and  diflference  are  given,  divide  the  differ- 
ence by  1  minus  the  rate. 

EXAM  PLE  S. 

1.  In  a  school  77  pupils  are  present,  which  is  87i^  of 
the  whole  number  on  the  roll ;  how  many  are  there  on  the 
roll?    ■  Ans.  77^.875  =  88. 

2.  A.,,  in  selling  goods  for  13,840,  clears  20%  on  their 
cost ;  what  was  the  cost  price  ? 

A71S.  $3,840  -^  1.20  =  $3,200. 

3.  A  man  sold  a  watch  for  $220,  which  was  20^  below 
its  value ;  what  was  its  value  ?    Ans.  $220  -^  .80  =  |275. 

4.  A  man  spends  $1,230  a  year,  which  is  82^  of  his 
salary ;  what  is  his  salary  ? 

5.  A  merchant  sold  cloth  at  $5.25  a  yard,  which  was  an 
advance  of  2h%  on  the  cost  price;  what  was  the  cost 
price  ? 

6.  The  population  of  a  town  is  18,558,  which  is  20^ 
greater  than  it  was  5  years  ago ;  what  was  it  then  ? 

7.  In  a  mixture  of  wine  and  water  there  are  12^  gals,  of 
water,  which  is  18J^  of  the  whole  ;  how  many  gallons 
in  all  ? 


PEECENTAGE.  197 

8.  A.'s  salary  is  $2,925,  which  is  66%  of  B/s ;  what  is 
B.'s  salary  ? 

9.  This  year  there  are  216  students  in  an  academy, 
which  is  20%  more  than  there  were  last  year ;  how  many 
were  there  then  ? 

10.  The  breadth  of  a  field  is  36^  rds.,  which  is  21!%  less 
than  its  length ;  what  is  its  length  ? 

11.  By  increasing  the  width  of  a  sheet  26%  it  was  made 
116  ft.  wide ;  how  wide  was  it  before  ? 

12.  A  farmer  sold  36%  of  his  sheep  at  $4J  each,  and 
received  for  them  $297.50  ;  how  many  sheep  had  he  ? 

MISCELLANEOUS    PROBLEMS    IN     PERCENTAGE. 

1.  What  is  the  difference  between  6^%  of  $800  and  Gi% 
of  $1,050  ? 

2.  A  farmer  raises  850  du.  of  wheat :  he  sells  18^^  of 
it  at  $1.25  a  bushel,  50^  of  it  at  $1|-  a  bushel,  and 
the  remainder  at  $1.75  a  bushel ;  what  does  he  get  for 
it  all  ? 

3.  What  is  the  difference  between  £2,971  and  37^^  of 
£2,971  ? 

4.  A  drover  sold  40  sheep  for  $248,  which  was  55^ 
advance  on  their  cost;  what  did  they  cost  him  apiece? 

5.  What  number  is  that  which,  being  diminished  by 
30^  of  itself,  gives  385? 

6.  A  market-woman  has  600  eggs,  and  a  second  market- 
woman  has  15^  more  ;  how  many  has  she  ? 

7.  A  young  man  spent  $18,750,  which  was  37^^  of  his 
inheritance  ;  how  much  did  he  inherit? 

8.  The  population  of  a  certain  town  in  1870  is  15,340, 


198  PERCENTAGE     AND     ITS     APPLICATIONS. 

which  is  an  increase  of  18  per  cent,  on  its  population  in 
1860 ;  what  was  the  population  in  1860  ? 

9.  The  distance  between  two  towns  in  France  is  20^ 
more  than  4  hilometers  ;  what  is  the  distance  ? 

10.  A  man  had  11  hectol  of  wine,  but  he  lost  d%  of  it 
by  leakage ;  how  much  had  he  left  ? 

11.  A  drover  sold  cows  and  sheep  for  19,180 ;  he  received 
for  his  sheep  70^  of  what  he  got  for  his  cows ;  what  did 
he  get  for  the  cows  ? 

12.  A  farmer  raises  wheat  and  corn ;  his  wheat  crop  is 
worth  $1,036,  which  is  40^  more  than  the  value  of  his 
corn  crop :  what  is  the  value  of  the  corn  crop  ? 

13.  From  a  cask  of  wine  37^  was  drawn  off  and  33.39 
gallons  remained ;  how  many  gallons  did  it  contain  ? 

14.  A.  invests  35^  of  his  capital  in  land  and  has  113,000 
remaining ;  what  is  his  capital  ? 

15.  An  army  loses  27^  of  its  number  in  battle  and  has 
22,630  men  remaining ;  how  many  did  it  contain  ? 

16.  A  man  bought  a  house  for  11,225.50,  which  in  3 
years  rose  in  value  147^ ;  what  was  it  then  worth  ? 

17.  A  man  had  15,420;  he  bought  goods  with  37^^  of 
it,  and  then  lent  25^  of  the  balance  to  a  friend :  how  much 
had  he  left  ? 

18.  A  general  had  an  army  of  10,816  men,  of  whom  he 
lost  in  action  6J^;  how  many  did  he  lose  ? 

19.  A  merchant  bought  15  pieces  of  cloth,  each  contain- 
ing 31  J-  yards,  and  found  on  examination  that  50^  was 
damaged ;  how  much  was  good  ? 

20.  A  man  bought  75  acres  of  land  at  I42|  an  acre  and 
sold  it  all  for  $3,5 77i ;  what  per  cent,  did  he  gain  ? 


COMMISSION.  199 

21.  A  man  has  a  capital  of  $12,500  :  he  puts  16%  of  it 
in  stocks,  ^S^fo  in  land,  and  25^  in  mortgages ;  how  many 
dollars  has  he  left  ? 

22.  Henry  Adam  bought  a  house  for  $7,520,  spent 
$4,220  on  it  for  repairs,  and  $75  for  other  expenses ;  he 
then  sold  it  for  $17,427.12^ ;  how  much  did  he  gain  per 
cent.  ? 

REVIKV7     QUESTIONS. 

(158.)  Define  per  cent.  (159.)  What  is  the  rate  per  cent.? 
(160.)  What  is  percentage  ?  (161.)  What  is  the  base  ?  (162.) 
The  amoant?  (163.)  The  difference?  (164.)  Repeat  the  5 
principles  of  percentage.  (165-168.)  Give  the  corresponding 
rules. 


II.    COM  M  ISSION. 

DEFINITIONS. 

169.  Commission  is  a  percentage  paid  to  an  agent 
for  transacting  business. 

170.  An  Agent  is  one  that  transacts  business  for 
another.  If  he  buys  and  sells  merchandise,  he  is  called  a 
Commission  Merchant,  or  Factor;  if  he  buys  and 
sells  stocks,  exchange,  real  estate,  and  the  like,  he  is  called 
a  Broker ;  if  he  collects  debts,  taxes,  and  the  like,  he  is 
called  a  Collector. 

171.  A  Consignment  is  a  quantity  of  merchandise 
sent  to  an  agent  for  sale.  The  party  that  sends  the  goods 
is  the  Consignor,  and  the  agent  that  receives  them  is  the 
Consignee. 

173.  An  Account  of  Sales  is  an  account  rendered 
by  the  Consignee  to  the  Consignor.     The  amount  due  the 


200  PERCEKTAGE    AND    ITS    APPLICATIOl^^S. 

consignor,  after  deducting  commission  and  other  expenses, 
is  called  the  net  proceeds. 

mS,  All  problems  in  Commission  are  solved  by  the 
rules  for  percentage. 

The  base  on  which  commission  is  reckoned  is  what  the  agent  ex- 
pends, or  collects,  on  account  of  his  principal ;  except  in  buying 
and  selling  stocks,  and  the  like,  where  the  commission  or  hrokerage, 
as  it  is  called,  is  usually  laaed  on  the  par  value. 

EXAMPLES. 

1.  A  factor  received  a  consignment  of  flour,  which  he 
sold  for  $3,750 ;  what  was  his  commission  at  ^%  f 

Explanation. — Here  the  lase  is  $3,750  and  the  rate  A.^%  ;  hence, 
the  percentage,  or  commission,  is  $3,750  x  .045  =  $168.75. 

2.  A  cotton  broker  sells  70  bales  of  cotton  for  $80  per 
bale ;  what  is  his  commission  at  Z%  ? 

3.  A  real  estate  broker  sells  a  house  for  $23,750,  at  a 
commission  of  1\%',  what  must  he  pay  his  principal? 

4.  A  drover  sells  cattle  for  $4,250,  at  a  commission  of 
^% ;  what  does  he  pay  the  owner  of  the  cattle  ? 

5.  A  consignee  sold  300  ihls.  of  flour  at  $7  per  barrel ; 
what  is  his  commission  at  2J^  ? 

6.  An  auctioneer  sold  a  house  and  furniture  for  $26,750 ; 
what  was  his  commission  at  1\%  ? 

7.  A.  sold  500  pieces  of  cloth  at  $30  a  piece  and  paid  the 
owner  $14,700 ;  what  was  the  rate  of  commission  ? 

8.  A  factor  sold  500  pieces  of  muslin,  each  containing 
21  yds.,  for  23  cents  a  yard ;  what  was  his  commission  at 

9.  A  real  estate  broker  bought  a  house  for  $21,300,  and, 
by  direction  of  his  principal,  sold  it  again  at  an  advance 


COMMISSION^.  201 

of  20^  on  the  cost;  what  was  his  total  commission  at  the 
rate  of  1^%  both  for  buying  and  for  selling  ? 

10.  A  commission  merchant  receives  13,825  to  invest  in 
flour  on  a  commission  of  2% ;  what  is  his  commission  ? 

Explanation.— Here  the  amount  is  $3,825  and  the  rate  2%  ; 
hence,  the  base  is  $3,825-^1.02  —  $3,750,  which  is  the  cost  of  the 
flour.    The  commission  is  $3,750  x  .02  =  $75. 

11.  A  merchant  in  New  York  sends  $12,600  to  his  fac- 
tor in  Chicago  to  buy  flour,  agreeing  to  pay  5^  of  the  cost 
for  commission ;  how  much  flour  does  he  receive,  the 
market  price  being  $12  a  barrel  ? 

12.  A  manufacturer  invested  $22,050  in  cotton;  the 
market  price  was  15  cts.  a  pound,  the  commission  2^%, 
the  freight  and  cartage  1^%,  and  the  insurance  H% ;  how 
many  pounds  did  he  buy  ? 

13.  A  commission  merchant  sold  120  pieces  of  muslin, 
each  containing  30  J  yds.,  at  22  cts.  a  yard,  at  a  commission 
of  5i% ;  how  much  did  he  receive  ? 

14.  A.  sold  75  firkins  of  butter,  each  weighing  56  ZJs., 
at  22^  cts.  a  pound ;  what  was  his  commission  at  6%  ? 

15.  An  auctioneer  sold  150  hhds.  of  sugar,  each  weigh- 
ing 1,150  lis.,  at  $7  a  hundred,  on  a  commission  of  1 J^ ; 
what  was  his  commission,  and  how  much  did  he  pay  to  the 
owner  ? 

16.  A  broker  receives  $3,500  to  buy  cotton  at  8  cts.  a 
pound,  his  commission  being  If  ^ ;  how  much  does  he  buy  ? 

17.  A  broker  bought  a  house,  charging  2^%  commission  ; 
his  bill  was  $3,224.24;  what  did  he  pay  for  the  house  ? 

18.  A  commission  merchant  sold  goods  for  $7,500  at 
3^%  commission ;  what  did  he  make  ? 


202 


PERCENTAGE    AND     ITS    APPLICATIONS. 


Find  the  net  proceeds  of  the  following  accounts  of  sales: 
(19.) 

S.  A.  Betts,  Romford,  Ct. 


Sales  on  account  of 


1878. 

BUYER. 

DESCRIPTION. 

e 

Ct8. 

Jan.      4. 

S.  T.  D. 

78  Jm.  oats. 

@65c. 

50 

70 

"       15. 

P.  Q.  R. 

115  bu.  corn. 

@93c. 

106 

95 

Feb.    11. 

R.  S.  V. 

1330  lbs.  butter, 

@  31c. 

409 

20 

March  5. 

P.  S.  Q. 

2560^65.  cheese. 

@14tf. 
unt  .    . 

358 

40 

Gross  amo 

925 

25 

Cha/rges. 

Freight  and  cartage 

Storage . 

Commission  on  $925.25  (^4^fo  . 

.    .    $39.54 
.    .        5.25 
.    .     41.63^ 

86 

•    •    • 

42i 

Net  proc€ 

jeds 

$838 

83^ 

Sales  on  account  of 


(30.) 
Jos.  p.  QuiNN,  Roxhury,  N,  Y. 


1873. 

BUTBK. 

DESCRIPTION. 

$ 

cts. 

June  13. 
July    5. 

C.  D. 

D.  L. 

S.  K 

1563  lbs.  pork,  @  10c.  .     .     . 
408  lbs.  ham,  @  19c.  .     .     . 
829  lbs.  lard,    @  14c.  .     .     . 

Freight  and  carl 
Commission  at  I 

Net  proce 

Gross  amount    .     .     . 

Cha/rges. 
tage.     ......      $5.42 

S% 17.491 

«d8 

$326 

961 

INSURANCE.  203 


REVIETAT     QUESTIONS. 

(169.)  What  is  commission?  (170.)  Wliat  is  an  agent?  A 
commission  merchant,  or  factor ?  A  broker?  A  collector?  (171.) 
What  is  a  consignment?  A  consignor?  A  consignee?  (172.) 
What  is  an  account  of  sales?  Net  proceeds?  (173.)  By  what 
rules  are  problems  in  commission  solved  ?    What  is  the  base  ? 


III.     INSURANCE. 

DEFINITIONS. 

174.  Insnrance  is  a  guarantee  of  indemnity  in  case  of 
loss  by  fire,  or  other  casualty. 

175.  A  Policy  of  Insurance  is  a  written  contract,  in 
which  a  company  agrees  to  pay  a  certain  sum  in  case  of 
loss. 

176.  The  Premium  is  a  percentage  paid  to  the  com- 
pany as  a  compensation  for  the  risk  assumed. 

Note. — There  are  various  kinds  of  insurance ;  as,  Fire  Insurance, 
Marine  Insurance,  Life  Insurance,  Accident  Insurance,  and  the 
like.  These  differ  from  each  other  in  the  nature  of  the  risk  assumed 
and  in  the  mode  of  determining  the  rate  'per  cent.  The  rate  per  cent. 
and  the  tirnes  of  paying  the  premium  having  been  fixed,  the  method 
of  proceeding  is  essentially  the  same  in  all. 

177.  All  problems  of  insurance  are  solved  by  the  rules 
for  percentage. 

The  has6  on  which  the  premium  is  reckoned  is  the  sum  named  in 
the  policy. 

EXAM  PLES. 

1.  What  premium  must  be  paid  to  insure  a  house  for 

$8,000  for  1  year  at  \%  ? 

Explanation.— Here,  the  Mse  is  $8,000  and  the  rate  f  %  ;  hence, 
the  percentage  or  premium  is  |8,000  x  .OOf  =  $50. 


204  PERCEN-TAGE     AKD     ITS    APPLICATIONS. 

2.  What  is  the  premium  for  insuring  a  ship  and  cargo 
valued  at  173,850  at- the  rate  of  3^^  ? 

3.  A.  insured  his  house  for  1  year  for  18,000  at  the  rate 
of  i%,  and  his  furniture  for  $3,000  at  the  rate  of  ^%  ;  what 
was  the  total  premium  ? 

4.  A  merchant  insures  his  store  for  $12,000  at  the  rate 
of  i%,  and  his  stock  of  goods  for  $15,000  at  the  rate  of 
1^%  ;  what  is  the  entire  premium  ? 

5.  B.  owns  f  of  a  cargo  of  goods  worth  125,000,  and 
insures  his  interest  at  2^%  ;  what  is  the  premium  ? 

6.  A  vessel  and  cargo  valued  at  $37,900  are  insured  at 
the  rate  of  3%  ;  what  is  the  premium  ? 

7.  A  shipping  merchant  sends  wheat  valued  at  $1,200, 
from  Chicago  to  New  York,  which  he  insures  at  the  rate 
of  1^%  ;  what  premium  does  he  pay  ? 

8.  A  man  insures  his  house  for  $10,000  at  ^%,  his  barn 
for  11,800  at  i%,  and  his  furniture  for  $3,000  at  1^%; 
what  premium  does  he  pay  ? 

9.  A  cargo  of  goods  valued  at  $48,000,  insured  at  IJ^, 
is  injured  to  the  amount  of  37^  of  its  value ;  what  must 
the  company  pay,  over  and  above  the  premium  ? 

10.  A  merchant  pays  $2,340  on  a  vessel  and  cargo,  the 
rate  being  4J^ ;  for  what  sum  is  he  insured  ? 

Explanation. — Here,  the  percentage  is  $2,340  and  the  rate  4^%; 
hence,  the  base,  or  the  sum  insured,  is  $2,340  -^  .045  =  $52,000. 

11.  A  man  pays  $87.50  for  the  insurance  of  house  at  ^%, 
and  $50  for  the  insurance  of  furniture  at  IJ^  ;  if  both  are 
destroyed  by  fire,  how  much  will  he  receive  ? 

12.  Shipped  5,000  hbls.  of  flour  worth  $10.50  a  barrel, 
and  paid  for  insurance  $2,887.50;  what  was  the  rate? 


INSURAKCE.  205 

Explanation.— Here,  the  base  is  $52,500  and  the  percentage 
$3287.50  ;  hence,  the  rate  is  $2887.50  -i-  $52,500  =  .055  =  5h%. 


13.  A  person  20  years  of  age  is  required  to  pay  1.; 
per  annum  to  insure  his  life ;  what  must  he  pay  each  year 
on  a  policy  of  $5,000? 

14.  A  man  40  years  of  age  wishes  to  insure  his  life  for 
5yrs.  and  finds  that  the  annual  rate  is  1.86^;  how  much 
must  he  pay  a  year  on  a  policy  of  $12,500  ? 

15.  What  does  it  cost  to  insure  a  cargo  of  goods  worth 
$50,000,  at  the  rate  of  1|^  ? 

16.  What  does  it  cost  to  insure  a  vessel  for  $13,000,  at 
2i%,  and  her  cargo  for  $18,268.50,  at  1^%  ? 

17.  What  is  the  premium  on  |  of  a  vessel,  worth  $27,500, 
at  3|^,  and  |  of  the  cargo,  worth  $126,875,  at  2%? 

18.  -A  merchant  shipped  400  bih.  of  fish,  worth  $3.50 

per  barrel;  for  what  amount  must  he  insure,  at  5%,  to 

cover  the  value  of  the  fish  and  the  cost  of  insurance  ? 

Explanation.— Here,  the  difference  is  $1,400  and  the  rate  5%; 
hence,  the  base  is  $1,400  -^  .95  =  $1,473. 68f. 

19.  I  have  goods  worth  $37,560.75,  which  I  insure  for  f 
of  their  value,  paying  $178.20 ;  what  is  the  rate  ? 

^ros.  -^^%. 

20.  A  merchant  insured  450  pieces  of  silk,  each  piece 
worth  $35J,  at  4|^;  what  was  the  premium  ? 

21.  I  insure  my  house  for  $8,000  for  3  years  at  ^%  per 
annum ;  what  is  the  total  premium  ? 

REVIEW      QUESTIONS. 

(174.)  What  is  insurance  ?  (175.)  What  is  a  policy  of  insur- 
ance? (176.)  What  is  the  premium?  Mention  some  of  the  dif- 
ferent kinds  of  insurance.  (177.)  By  what  rules  are  the  problems 
of  insurance  solved  ? 


206  PERCENTAGE     AND    ITS     APPLICATIONS. 

IV.     PROFIT     AND     LOSS. 

DEFINITIONS. 

178.  Profit  and  Loss  are  commercial  terms  indicating 

gain,  or  loss,  in  business  transactions. 

If  the  selling  price  of  any  article  is  greater  than  the  cost  price  there 
is  a  profit;  if  the  selling  price  is  less  than  the  cost  price  there  is  a 
loss.  Both  profit  and  loss  are  usually  reckoned  as  percentages  on  the 
cost  price,  as  a  hose. 

179.  The  problems  in  profit  and  loss  are  solved  by  the 
rules  for  percentage. 

EXAMPLES. 

1.  A  merchant  sold  goods  that  cost  $3,350,   at    an 

advance  of  20^  ;  what  was  his  profit  ? 

Explanation. — Here,  the  base  is  $2,350  and  the  rate  30%  ; 
hence,  the  percentage  or  profit  is  $3,350  x  .30  =  $470. 

2.  A  person  entered  into  a  speculation  in  which  he  in- 
vested $7,000,  and  cleared  lb% ;  what  was  his  profit  ? 

3.  Bought  a  horse  for  $325,  and  sold  him  again  at  an 
advance  of  1^%',  what  was  the  profit  ? 

4.  A  merchant  buys  cloth  at  16  per  yard,  and  sells  it 

again  so  as  to  clear  25^ ;  what  does  he  ask  per  yard  ? 

Explanation. — Here,  the  base  is  $6  and  the  rate  35%  ;  hence, 
the  amount,  or  selling  price,  is  $6  x  1.35  =  $7.50. 

5.  A  person  begins  business  with  a  capital  of  $18,000, 
and  in  one  year  increases  it  \%% ;  what  is  his  capital  at 
the  beginning  of  the  second  year  ? 

6.  Bought  1,280  Z^5.  of  sugar,  at  l^ds.  a  pound,  and 
sold  it  at  an  advance  of  25^ ;  what  was  the  profit  ? 

7.  Bought  drugs  for  £150,  and  sold  them  at  a  profit  of 
250^ ;  how  much  was  the  profit  ? 


PROFIT     Al^D     LOSS.  .  207 

8.  A  merchant  had  goods  worth  13,750,  of  which  66| 
per  cent,  were  destroyed  by  fire ;  what  was  his  loss  ? 

9.  Bought  a  house  for  124,500,  expended  $1,500  in  re- 
pairs, $4,000  for  furniture,  and  $800  for  taxes ;  what  must 
the  whole  be  sold  for,  to  get  an  advance  of '14^? 

10.  Bought  a  1000  T.  of  coal,  at  $5  a  ton,  and  sold  the 
same  at  a  profit  otll%;  what  was  the  entire  gain  ? 

11.  Bought  butter  at  18  cts.  a  pound,  and  lost  by  the 
purchase  33 J^  per  cent;  what  was  the  selling  price  ? 

12.  Bought  a  I  of  a  factory,  and  then  sold  -J  of  my  share 
for  $15,000,  making  50^  on  the  cost ;  what  was  the  value 
of  the  whole  factory  at  the  time  of  purchase  ? 

13.  Sold  goods  at  a  loss  of  8%  on  their  cost,  and  received 
$8,280  ;  what  did  they  cost  ? 

14.  Sold  a  horse  for  $364,  gaining  12^  on  its  cost;  what 
did  it  cost  ? 

15.  If  I  sell  a  horse  for  $240,  and  lose  20^,  what  should 
I  have  sold  him  for  to  gain  10^  ? 

16.  A.  hires  a  piece  of  ground  for  $120,  and  spends  $625 
for  Durham  calves,  and  $250  for  incidental  expenses ;  how 
much  must  he  get  per  head  for  the  calves  to  gain  20^  ? 

17.  A  stationer  sold  quills  at  $3.75  a  thousand,  and 
cleared  25^  on  their  cost;  how  many  per  cent,  would  he 
have  cleared,  if  he  had  sold  them  at  $4.50  a  thousand  ? 

18.  A.  bought  a  factory  for  $8,000,  and  stocked  it  at  a 
cost  of  $13,500 :  the  building  was  destroyed  by  fire,  but 
60^  of  the  stock  was  saved ;  what  was  his  loss  ? 

19.  A  grocer  bought  500  bags  of  coffee,  each  bag  con- 
taining 49  J  lbs.,  at  12  cts.  a  pound,  and  sold  it  at  a  profit  of 
16|5^ ;  for  what  did  he  sell  it  ? 


208  PERCENTAGE     AND     ITS     APPLICATIONS. 

20.  A  dealer  bought  375  T.  15  cwt.  of  wool,  at  175  a  ton, 
and  sold  it  at  a  profit  of  20^ ;  what  did  he  gain  ? 

21.  A  grocer  bought  410  Jw.  of  potatoes,  at  96  c^5.  a 
bushel,  and  sold  them  for  $492 ;  what  per  cent,  did  he 
make  ? 

22.  A  merchant  sold  goods  for  $900,  losing  10^ ;  what 
did  they  cost  him  ? 

23.  A  dealer  sold  20  ills,  of  flour  for  $7^  per  barrel,  and 
made  25^;  what  did  all  the  flour  cost  him  ? 

24.  A  person  laid  in  25  tons  of  coal  during  the  summer, 
and  saved  thereby  25^^  on  the  winter  price,  which  was 
$7.20  per  ton ;  what  did  his  coal  cost  him  ? 

25.  A  merchant  bought "250  hhls.  of  floup,  at  S5.50  per  hhh, 
and  sold  it  for  $1,875 ;  what  per  cent,  did  he  make  ? 

26.  A  grocer  sold  coffee  at  40  cts.  a  pound,  and  cleared 
25^ ;  how  should  he  have  sold  it  to  clear  12-|-^  ? 

27.  A.  sold  an  engine  for  $24,000,  and  lost  4^;  how 
much  would  he  have  received  if  he  had  made  10^? 

28.  A  man  sold  a  pair  of  horses  for  $224  and  gained 
40^  by  the  transaction ;  what  was  their  cost  ? 

29.  If  flour  at  $9  a  barrel  gives  a  profit  of  20^,  what 
does  it  cost  a  barrel  ? 

30.  Bought  books  for  $420,  and  sold  them  for  $357; 
how  many  per  cent,  did  I  lose  ? 

31.  A  cask  containing  S6 gals,  lost  6^ gals,  by  leakage; 
what  was  the  loss  per  cent.  ? 

32.  Mr.  Johnson  sold  a  carnage  for  $338.40  and  gained 
20^ ;  what  did  he  pay  for  it  ? 

.  33.  Bought  muslin  at  10  cts.  a  yard,  and  sold  it  at 
12^-  cts.  a  yard ;  what  per  cent,  did  I  gain  ? 


TAXES.  209 

34.  A  flock  of  sheep  increases  from  88  to  110  in  a  year ; 
what  is  the  gain  per  cent.  ? 

35.  A  grocer  sold  potatoes  for  116.10,  gaining  16% ;  if 
he  had  sold  them  for  $18.20,  how  much  would  he  have 
made  above  the  cost  price  ? 

REVIE^A^     QUESTIONS. 

(178.)  What  are  profit  and  loss?  When  is  there  a  profit  in  busi- 
ness? Wlien  a  loss?  What  is  the  base?  (179.)  By  what  rules 
do  we  solve  problems  in  profit  and  loss  ? 


V.     TAXES. 

DEFINITION. 

180.  A  Tax  is  a  sum  of  money  levied  on  a  commu- 
nity for  the  support  of  government,  or  for  public  improve- 
ments. 

The  United  States  government  is  supported  by  taxes  on  imported 
goods,  by  taxes  on  certain  manufactures,  by  sales  of  public  lands, 
stamps,  and  the  like. 

Money  for  the  support  of  state,  city,  county,  and  other  subordinate 
branches  of  government  is  generally  raised  by  a  tax  on  property. 
In  some  of  the  States,  however,  a  small  personal  tax  is  laid  on  each 
male  citizen  over  21  years  of  age  ;  this  is  called  a  poll-tax,  and  each 
person  so  taxed  is  called  a  poll. 

TAX  ON  PROPERTY  AND  POLLS. 

181.  A  Tax  on  Property  is  a  percentage  based  on 

the  Assessed  Value  of  the  property;   that  is,  on  its 

value  as  determined  by  officers  appointed  for  the  purpose, 

called-  Assessors. 

Property  is  of  two  kinds,  real  and  personal.  Real  estate  is  fixed 
property,  as  houses  and  lands ;  personal  property  is  movable  prop- 
erty, as  money,  bonds,  cattle,  furniture,  and  the  like.  Both  real  and 
personal  property  are  liable  to  taxation. 


210  PEECENTAGE     AND     ITS     APPLICATIOKS. 

METHOD    OF    LAYING    A    TAX. 

183.  The  assessors  prepare  a  list  of  all  the  taxable 
property,  specifying  the  amount  belonging  to  each  prop- 
erty-holder within  the  district  to  be  taxed,  and  also  a  list 
of  polls.  The  aggregate  valuation  of  all  the  taxable  prop- 
erty is  the  Mse  of  taxation.  The  entire  sum  to  be  raised 
on  property  divided  by  this  base  gives  the  Rate  of  Tax- 
ation. The  amount  of  taxable  property  held  by  any  in- 
dividual, multiplied  by  the  rate,  gives  the  amount  of  his 
tax  on  property  ;  this,  increased  by  his  poll  tax,  gives  his 
total  tax. 

EXAMPLES. 

1.  A  town  is  to  be  taxed  $23,200,  on  an  assessed  valua- 
tion of  $2,900,000;  what  is  A.'s  tax  on  an  assessed  valua- 
tion of  114,275  ? 

Explanation. — Here  there  is  no  poll  tax  ;  consequently  the  rate 
is  $23,200  -V-  $2,900,000  =  .008 ;  hence,  A.'s  tax  is  $14,275  x  .008  = 
$114.20.  Ans. 

2.  A  town  is  to  be  taxed  $13,848 ;  its  assessed  valuation 
is  $1,452,000,  and  it  contains  390  polls  each  liable  to  a  tax 
of  $2 :  now  if  A.  is  liable  for  2  polls  and  an  assessed  val- 
uation of  $7,820,  what  is  his  tax  ? 

Explanation. — Here  the  poll  tax  is  $780,  and  this  taken  from 
$13,848  gives  $13,068  to  be  raised  on  property.  The  rate  is  there- 
fore $13,068  -^  $1,452,000  =  .009 ;  hence,  A.'s  property  tax  is 
$7,820  X  .009  =  $70.38,  and  this  increased  by  a  poll  tax  of  $4,  gives 
$74.38.  Ans. 

3.  In  the  same  town,  B.  is  liable  for  2  polls  and  is 

assessed  for  $8,290 ;  what  is  his  tax  ? 

4.  In  a  school  district,  a  tax  of  $375  is  levied  for  the 
support  of  schools ;  what  is  A.'s  tax  on  a  valuation  of 
$8,000,  the  entire  valuation  of  the  district  being  $120,000? 


SIMPLE     INTEREST.  211 

REVIEW      QUESTIONS. 

(180.)  What  is  a  tax  ?  What  is  a  poll  tax  ?  What  are  polls  ? 
(181.)  What  is  a  tax  on  property?  Describe  real  and  personal 
property.  (182.)  What  is  the  method  of  levying  a  tax  on  property 
alone  ?  Wliat  is  the  method  when  polls  are  counted  ?  How  is  the 
rate  determined  ? 


VI.     SIMPLE     INTEREST. 

DEFI  N  ITi  ONS. 

183.  Interest  is  a  percentage  paid  for  the  use  of 

money. 

Interest  is  reckoned  at  a  certain  rate  per  cent,  for  each  year. 

184.  The  Principal  is  the  sum  on  which  interest  is 
reckoned ;  the  Rate  is  the  per  cent,  per  annum  which  is 
to  be  paid ;  and  the  Amount  is  the  sum  of  the  principal 
and  interest  for  a  given  time. 

The  rate  of  interest  fixed  by  law  is  called  the  Legal 
Rate ;  interest  reckoned  at  a  greater  rate  is  called 
Usury. 

In  about  half  the  States  of  the  Union  the  legal  rate  is  6% ,  and  in 
about  half  of  the  remaining  ones  it  is  7%.  In  some  States  two  rates 
are  fixed,  between  which  any  rate  is  legal  if  agreed  to  by  the  par- 
ties. 

In  reckoning  time,  a  month  is  usually  regarded  as  ^  of  a  year, 
without  reference  to  the  number  of  days  it  may  contain,  and  a  day 
is  regarded  as  3V  of  a  month. 

185.  Simple  Interest  is  interest  reckoned  on  the 
principal  only.  Compound  Interest  is  interest  reck- 
oned on  the  principal  and  also  on  the  accrued  interest  as 
it  falls  due. 

Note. — The  rules  for  interest  are  but  special  applications  of  the 
general  rules  of  percentage. 


212  PERCENTAGE     AND     ITS     APPLICATIOKS. 

MENTAL     EXERCISES. 

1.  What  is  the  interest  on  $15  for  1  year  at  6%  per 
annum  ?  What  is  the  principal  in  this  case  ?  The  rate  ? 
The  amount  ?  What  is  the  interest  on  $25  for  1  year  at 
S%?  What  is  the  principal  here?  The  rate?  The 
amount  ? 

2.  What  is  the  interest  on  $30  at  11%  for  1  year?  for  2 
years  ?  for  4  years  ?  for  9  years  ?  In  the  last  case,  what  is 
the  principal  ?  The  rate  ?  The  interest  ?  The  amount  ? 
The  time  ? 

3.  What  is  the  interest  on  $16  for  5  years  at  6%  ?  For 
3  years  at  7%  ?  For  4  years  at  10^  ?  On  $20  for  3  years 
at  6%  ?    On  $50  for  2  years  at  9^  ? 

186.  To  find  the  Interest  when  we  know  the 
Principal,  the  Rate,  and  the  Time  in  Years. 
Let  it  be  required  to  find  the  interest  on  $860  for  3  yrs, 

at   6%.  OPEBATION. 

Explanation.— The  interest  on  $860  for  lyr.  is  ^^^^ 

equal  to  $860  x  .06,  or  to  $51.60  ;  for  Syrs.  it  is  3  _ 

times  as  great,  or  $51.60  x  3  =  $15480.  $51.60 

In  like  manner  all  similar  cases  may  be  treated,  3 

hence  the  following  $154  80 

RULE. 

Multiply  the  principal  hy  the  rate,  and  that 
result  by  the  time  in  years. 

Note.— In  applying  this  and  the  following  rules,  let  all  decimals 
be  carried  to  three  places,  adding  1  to  the  last  figure  when  the  first 
figure  rejected  is  equal  to  or  greater  than  5. 

EXAMPLES. 

1.  What  is  the  interest  on  $560  for  4  years  at  the  rate  of 
7%  per  annum  ?  Ans.  $560  x  .07  x  4  =  $156.80. 


SIMPLE   INTEREST.  213 

2.  What  is  the  interest  on  $794  for  Siyrs.  at  7^  ? 

3.  Find  the  interest  on  $8,942  for  3  J  years  at  S^%. 

4.  Compute  the  interest  on  $8,720  for  1^  years  at  7^. 

5.  Find  the  interest  on  $712  for  3  years  at  6^. 

6.  What  is  the  interest  on  $329.50  for  2  years  at  7^  ? 

7.  What  is  the  interest  on  $986.30  for  1  year  at  6^%  ? 

8.  Find  the  interest  on  $12,600  for  1  year  at  4rl%. 

9.  Find  the  interest  on  $112.75  for  4:  yrs.  at  10^. 

10.  Find  the  interest  on  $2,884.25  for  3  yrs.  at  6%. 

11.  Find  the  interest  on  $1,750  for  2  yrs.  at  Q%. 

12.  Find  the  interest  on  $396.50  for  5  yrs.  at  S%. 

To  find  the  Amount,  add  the  interest  to  the  prin- 
cipal. 

13.  To  what  will  $1,400  amount  in  2  years  at  the  rate 
of  3i%  per  annum  ?  A^is.  $1,400  +  $98  =  $1,498. 

14.  Find  the  amount  of  $4,186.25  for  1  year  at  6^%. 

15.  What  is  the  amount  of  £168  for  1  year  at  7^  ? 

16.  Find  the  amount  of  $1,001.75  for  1  yr.  at  6|^. 

Ans.  $1,068,116. 

17.  What  does  $450  amount  to  in  2|  years  at  6%  ? 

18.  Find  the  amount  of  $3,875.20  for  5  years  at  4f ^. 

19.  What  does  £2,000  amount  to  in  7  years  at  S^%  ? 

20.  To  what  will  $7,500  amount  in  5  yrs.  at  3^%  ? 

21.  Find  the  amount  of  $736  for  4:  yrs.  at  10^. 

22.  Find  the  amount  of  $1,490  for  Si  yrs.  at  12^. 

23.  Find  the  amount  of  $2,714  for  5  yrs.  at  6%. 

24.  Find  the  amount  of  $10,863^  for  1  yr.  at  8^. 

25.  Find  the  amount  of  $16,314  for  2  yrs.  at  7%, 

26.  Find  the  interest  on  $9,000  for  2  yrs.  at  6%, 

27.  Find  the  interest  on  $4,000  for  3  yrs.  at  10^. 


214  PERCEl^TAGE     AND     ITS     APPLICATION'S. 

187.  To  find  the  Interest  when  we  know  the 
Principal,  the  Rate,  and  the  Time  in  Months. 

Let  it  be  requred  to  find  the  interest  on  $480  for 
9  months  at  the  rate  of  1%  per  annum. 

Explanation.— The  interest  on  $480  for  1  1480 

year  is  $33.60  ;  hence,  the  interest  on  the  same  .07 

sum  for  1  month  is  jV  of  $33.60,  or   $2.80;  l^'liSS  60 

consequently,  the  interest  for  9  months  is  9  ^aio'cA 

times  as  great,  or  $25.20.  ^^'^^ 

In  like  manner  we  may  treat  all  similar  1 

cases  ;  hence,  the  following  Ans.   $25.20 

RULE. 

Multiply  the  principal  hy  the  rate  and  divide 

the  product  hy  12 ;   then  multiply  the  quotient  hy 

the  nujnher  of  months. 

Note. — The  time  may  be  reduced  to  decimals  of  a  year,  and  then 
the  last  rule  can  be  used ;  but  in  most  cases,  and  especially  when 
the  rate  is  6%,  the  rule  here  given  is  the  simpler. 

EXAM  PLES. 

1.  What  is  the  interest  on  $815  for  11  mos,  at  the  rate 
of  7%  per  annum  ? 

Ans.  $815  X  .07  -^  12  x  11  =  152.296. 

2.  What  is  the  amount  of  $1,375  for  4mos.  at  the  rate 
of  5^  per  annum  ? 

Ans.  11,375  +  122.917  =  11,397.917. 

3.  Find  the  interest  on  $1,742.10  for  7  mos.  at  6^%. 

4.  Find  the  interest  on  $840  for  9  7nos.  at  4J^. 

5.  Find  the  interest  on  $711  for  14  7nos.  at  8%. 

6.  Find  the  amount  of  $1,285  for  16  mos.  at  G%. 

7.  Find  the  interest  on  $748  for  8  mos.  at  6%. 

Ans.  $29.92. 


SIMPLE     INTEREST.  215 

8.  Find  the  amount  of  14,316  for  9  mos.  at  10^. 

9.  Find  the  amount  of  12,872  for  7  mos.  at  b%. 

10.  Find  the  interest  on  1911.50  for  3  mos.  at  8^. 

11.  Find  the  interest  on  179.48  for  10  mos,  at  4^. 

12.  Find  the  amount  of  $693.25  for  9  mos.  at  t%, 

13.  Find  the  interest  on  1748  for  8  mos.  at  6^. 

Simplification. — When  the  rate  per  annum  is  6%,  the  rate  per 
month  is  1% ,  or,  .005.  In  this  case  we  multiply  the  principal  by  half 
the  number  of  months  and  then  divide  by  100,  or  what  is  the  same 
thing,  we  divide  half  the  number  of  months  by  100  and  multiply  the 
principal  by  the  resulting  quotient.  Thus,  in  example  13,  we  have 
$748  X  .04  =  $29.92.    Ans. 

14.  What  is  the  interest  of  $890  for  10  months  at  6^  ? 

15.  Find  the  interest  on  $1,175  for  14  months  at  6^. 

16.  Find  the  interest  on  $8,742.75  for  9  mos.  at  Q%. 

17.  Find  the  interest  on  $846  for  15  months  at  6^. 

18.  What  is  the  interest  on  $750  for  14^  months  at  6^  ? 

19.  Find  the  amount  of  $872  for  10  months  at  Q%. 

20.  Find  the  amount  of  $942  for  15  mos.  at  6^. 

21.  Find  the  interest  on  $1,796  for  17  mos.  at  Q%. 

Note. — If  the  time  is  given  in  months  and  days,  or  in  yea/rs, 
months  and  days,  reduce  it  to  months  and  decimals  of  a  month  and 
proceed  as  before. 

The  operation  of  reduction  can  always  be  performed  mentally. 

22.  What  is  the  interest  on  $480  for  3  yrs.  4  mos.  21  da. 
at  the  rate  of  7^  ? 

Expi^NATiON. — Here  we  see  that  ^yrs.  ^^ttios.  is  40wos.,  and 
tliat  21  da.  is  f  ^  or  .7  of  a  month ;  hence,  the  time  in  months  is 
40.7  mos.  Finding  the  interest  on  $480  for  1  month,  we  have  $2.80 ; 
hence,  $2.80  x  40.7  =  $113.96.    Ans. 

Find  the  interest 

23.  On  $1,640  for  4  years,  5  months,  and  12  days  at  7^. 


21G  PERCENTAGE     AKD     ITS     APPLICATIONS. 

24.  On  12,306  for  1  year,  7  months,  27  days,  at  5%- 

25.  On  $1,260  for  3  years  and  6  days  at  1%. 

26.  On  $1,620  for  5  yrs.  24  da.  at  4=%. 

27.  On  $675.89  for  3  yrs.  6  mos.  6  da.  at  8^. 

28.  On  $864,768  for  9  mos.  25  da.  %i%. 

29.  On  $100  for  1  yr.  3  mos.  10  67«.  at  6^. 

30.  On  $1,000  for  9  mos.  15  ^«.  at  1%. 

31.  On  $1,700  for  %yrs.  dmos.  10  da.  at  5^ 

32.  On  £2,500  for  5  mos.  18  da.  at  4^%:. 

33.  On  $450  for  3  yrs.  6  mos.  18  ^a.  at  6%- 

34.  On  $710  tor  3  yrs.  10  mos.  at  7^. 

35.  On  $1,766  forlyr.  4:  mos.  IS  da.  at  6%. 

Explanation.— Here  the  rate  is  6%  ;  hence,  we  di\ide  half  the 
number  of  months  by  100,  which  gives  .083,  md  multiply  the  prin- 
cipal by  this  result.     Thus,  $1,766  x  .083  =  $146,578.     Aiis. 

30.  What  is  the  interest  on  $14.50  for  19  days  at  6%? 
Ans.   $14.50  X  .3167  =  $0.0459. 

37.  The  amount  of  £10,000  for  3  yrs.  7  mos.  12  da.  at  6^? 

38.  What  is  the  interest  of  $2,300  from  May  3d,  1870,  to 
January  15th,  1873,  at  Q% ? 

39.  What  is  the  amount  of  $3,150  from  August  16th, 
1861,  to  May  1st,  1869,  at  %%  ? 

40.  The  amount  of  $5,675  ior^yrs.  9  mos.  24=  da.  at  6%  ? 

41.  The  interest  on  $3,000  for  4:  yrs.  8  mos.  6  da.  at  6%  ? 

42.  The  interest  on  $2,500  for  lyr.  %mos.  12  da.  at  7^? 

43.  The  amount  of  $3,500  for  2 yrs.  4  mos.  18  da.  at  4^%  ? 

44.  The  amount  of  $850  for  1  yr.  9  7nos.  15  da.  at  7i%  ? 

45.  The  interest  on  $1,800  for  2 yrs.  3  ?nos.  10 da.  at  8%? 

46.  The  interest  on  $2,100  for  9  mos.  12  da.  at  ^%  ? 

47.  The  interest  on  $1,100  for  llanos.  Qda.  at  7%? 


SIMPLE     INTEREST.  217 

48.  The  interest  on  $3,000  for  3  yrs.  6  mos.  18  da,  at  b%  ? 

49.  The  amount  of  $4,000  for  1  yr.  3  7nos.  12  da,  at  4=%  ? 
60.  The  amount  of  $5,000  for  lyr.  Imo.  Qda.  at  7%? 

188.  To  find  the  Interest  for  Days  at  the  Rate 
of  6^. 

Explanation. — In  business  transactions,  interest  for  days  is  com- 
puted on  the  supposition  that  30  days  make  1  month,  and  12  months 
1  year,  that  is,  that  the  year  consists  of  360  days.  In  this  case, 
when  the  rate  is  6  ^ ,  the  interest  on  $1  for  6  days  is  1  mill ;  hence, 
the  following 

RULE. 

Multiply  the  pj%ncipal  by  the  nurriber  of  days, 
divide  the  result  by  6,  and  then  move  the  decimal 
point  three  places  to  the  left 

EXAM  PLES. 

1.  What  is  the  interest  on  184.60  for  15  da.  at  Q%  ? 

Solution.— We  multiply  $84.60  by  15  and  divide  the  result  first 
by  6  and  then  by  1,000  ;  hence,  $86.40  x  15  -f-  6,000  =  $.213.     Ana. 
The  operation  can  often  be  simplified  by  cancellation. 

2.  What  is  the  interest  on  1175.20  for  18  days  at  6^  ? 

3.  What  is  the  amount  of  $144  for  25  days  at  6^  ? 

4.  What  is  the  interest  on  $710  for  11  da.  at  6^  ? 

5.  On  1334.56  for  13  da.  at  6^? 

6.  On  $511.27  for  17  da.  at  Q%  ? 

7.  On  $3,942.75  for  18  da.  at  6%  ? 

Note.— Having  found  the  interest  for  days  at  the  rate  of  6%,  we 
may  find  the  interest  at  other  rates  by  the  method  of  aliquot  parts. 
Thus,  to  find  the  interest  at  5  % ,  we  diminish  the  interest  at  6  %  by 
its  sixth  part ;  to  find  the  interest  at  7%,  we  increase  the  interest  at 
6fohy  its  sixth  part.  The  interest  at  4|^%  and  at  7^%  are  respect- 
ively equal  to  the  interest  at  6  %,  diminished  and  increased  by  its 


218  PERCENTAGE     AND     ITS    APPLICATIONS. 

fourth  part.  The  interest  at  4%  and  at  8%  are  respectively  equal 
to  the  interest  at  6%,  diminished  and  increased  by  its  third  part, 
and  so  on. 

8.  Find  the  interest  on  I960  for  2^  da.  at  h%  and  at  1%. 

Solution.— The  interest  at  6%  is  $3.84  and  i^  of  this  is  $.64; 
hence,  at  Sf^  the  interest  is  $3.84  —  $.64  =:  $3.20,  and  at  7%  it  is 

$3.84 +  $.64  =  $448. 

9.  Find  the  interest  on  $1,230  for  84  da.  at  b%  and 
at  7^.  Ans.   $14.35  and  $20.09. 

10.  On  $960  for  QQ  days  at  ^%  and  at  7^^. 

11.  On  $648  for  54  da.  at  4^%  and  at  ^%, 

12.  On  $362.50  for  27  da.  at  4^  and  at  8^. 

13.  On  $187.75  for  90  da.  at  3^  and  at  9^. 

14.  On  $124.20  for  63  da.  at  b%  and  at  7^. 

189.   To  find  Accurate  Interest  for  Days. 

Explanation. — The  preceding  method  gives  a  result  too  great 
by  its  yV  P^^  J  *o  fii^d  the  accurate  interest,  we  may  diminish  the 
result  found  in  accordance  with  the  preceding  rule  by  its  -^\  part,  or 
we  may  use  the  following 

RULE. 
Find  the  interest  for   1   year  at  the  given  rate ; 
then  multiply  the  result  by  the  number  of  days 
and  divide  the  product  by  365. 

EXAMPLES. 

1.  What  is  the  accurate  interest  on  $803  for  35  days  at 
the  rate  of  7^  per  annum  ? 

Solution.— The  interest  of  $803  for  1  year  at  7^  is  $56.21 '; 
hence,  $56.21  x  35  -f-  365  =  $5.39.    Ans. 

2.  Find  the  accurate  interest  on  $584  for  70  da.  at  7^. 


SIMPLE     INTEREST.  219 

3.  On  $876  for  105  da.  at  8^  per  annum. 

4.  On  $3,712.25  for  93  da.  at  7^. 

5.  On  $112.70  for  63  da.  at  6%. 

6.  On  $396.64  for  63  da.  at  %. 

7.  On  $1,815  for  93  da.  at  5^. 

190.  To  find  the  Rate  when  we  know  the 
Principal,  the  Interest,  and  the  Time  in  Years. 

Let  it  be  required  to  find  the  rate  when  the  principal  is 
$712,  the  interest  $128.16,  and  the  time  3  yrs. 

Explanation.— The  inter-  operation. 

est  on  $712  for  ?>yrs.  at  1%  is         1712  X  .01  X  3  =  $21.36. 
$21.86  ;  but  the  given  interest 

is  $128.16,  that  is,  it  is  6  times  $128.16  -j-  $21.36  =  6^. 

as  great ;  hence,  the  required 
rate  is  six  times  1  % ,  or  6  % . 

In  like  manner  all  similar  cases  may  be  treated ;  hence,  the  fol- 
lowing 

RULE. 

Find  the  interest  at  1%  on  the  principal  for  the 
given  time;  then  divide  the  given  interest  by  the 
result. 

EXAMPLES. 

What  is  the  rate  per  annum  when 

1.  The  interest  on  $950  for  IQmos.  is  $88.66|? 

2.  The  interest  on  $380  for  1  yr.  4  mos.  is  $22.80  ? 

3.  The  interest  on  $8,726  for  l^yrs.  is  $916.23  ? 

4.  The  interest  on  $712  for  3  yrs.  is  $128.16  ? 

5.  The  interest  on  $329.5  for  2  yrs.  is  $46.13  ? 

6.  The  interest  on  794  for  S^yrs.  is  $194.53  ? 

7.  The  interest  on  $450  for  3  yrs.  6  mos.  18  da.  is 
$79.87i-? 


220  PERCENTAGE     AND     ITS     APPLICATIONS. 

191.  To  find  the  Time  when  the  Principal,  the 
Rate,  and  the  Interest  are  given. 

Let  it  be  required  to  find  the  time  iii  which  the  interest 
on  11,200  at  6%  will  he  equal  to  $120. 

Explanation.— The  in-  operation. 

terest  on  $1,200  for  1  year  il200  X    06  =  $72 

is  $72.      But  $72  is  con- 

tained  in  the  given  interest  !^  ^  ^  2  y^^^^  =  1  yr.  8  mOS, 

$120,    If    times  ;     conse-  $72  ^  ^  ^ 

quently,  the  required  time 
is  If  times  1  year  ;  that  is,  it  is  1  yr.  8  mo. 

In  like  manner  we  may  treat  all  similar  cases ;  hence,  the  fol- 
lowing 

RULE. 

Find  the  interest  on  the  pidiicipal  for  1  year  at 

the  given  rate ;    then  divide  the  given  interest  by 

the  result. 

EXAM  PLES. 

Find  the  time  in  which  the  interest 

1.  On  $712  at  6%  will  be  $128.16. 

2.  On  $8,942  at  Si%  will  be  $2,470.22J. 

3.  On  $329.50  at  7^  will  be  $46.13. 

4.  On  $980  at  6%  will  be  $44.10. 
6.  On  $1,175  at  6%  will  be  $82.25. 

6.  On  $846  at  6%  will  be  $63.45. 

7.  On  $872  at  6%  will  be  $915.60. 

8.  On  $1,500  at  1%  will  be  $210. 

9.  On  $3,000  at  6%  will  be  $600. 

192.  To  find  the  Principal,  when  the  Interest, 
the  Rate,  and  the  Time  are  given. 

Let  it  be  required  to  find  the  principal  that  will  give 
$65  interest  in  20  mos.  at  6%  per  annum. 


SIMPLE     INTEREST.  221 


OPBBATION. 


Explanation. — The  interest  on 

$1   for  20  mos.  is  10  ds. ;    now   if 

$1  draws  IQcts.  in  the  given  time,        |1  X  .06  X  If  =  10.10 

it  will  require  $650  to  draw  $65  in  |g5 

$65      .  xTry  =  $650. 

the  same  time :    hence,    zttt-t-    is  »U.i 

10  c^s.,  ^ 

equal  to  the  number  of  dollars  in  •*•  1650.   Ans. 

the  required  principal. 

In  like  manner  we  may  reason  on  all  similar  cases  ;  hence  the 

following 

RULE. 

Find  the  interest  on  $1  for  the  given  time  at 
the  given  rate  ;  then  divide  the  given  interest  by 
the  result. 

What  is  the  principal  on  which  the  interest 

1.  At  b%  for  l^mos.  is  $157.50  ? 

2.  At  6^  for  %yrs.  6  mos.  is  1450  ? 

3.  At  44^  for  dyrs.  4:  mos.  is  $412.50? 

4.  At  S%  for  27  mos.  is  $324  ? 

5.  At  7%  for  20.4: mos.  is  $297.50  ? 

6.  At  6%  for  15^^05.  is  $7.66|? 

MISCELLANEOUS      EXAMPLES. 

1.  The  interest  on  a  certain  sum  for  4  years,  at  7  per 
cent.,  is  $266 ;  what  is  the  principal  ? 

2.  The  interest  on  $3,675,  for  3  years,  is  $771.75  ;  what 
is  the  rate  ? 

3.  The  principal  is  $459,  the  interest  $183.60,  and  the 
rate  8  per  cent. ;  what  is  the  time  ? 

4.  The  interest  on  a  certain  sum  for  3  years,  at  6  per 
cent.,  is  $40.50  ;  what  is  the  principal  ? 

5.  The  principal  is  $918,  the  interest  $269.28,  and  the 
rate  4  per  cent. ;  what  is  the  time  ? 


222  PERCENTAGE     AND     ITS     APPLICATIONS. 

6.  What  sum  of  money  must  be  placed  at  interest  at  7^, 
for  3yrs.  9  mos.,  that  the  interest  may  be  1393.75  ? 

7.  In  what  time,  at  7  per  cent.,  will  a  mortgage  of 
$8,000,  whose  interest  is  unpaid,  amount  to  19,120  ? 

8.  If  I  buy  a  house  for  15,620  and  receive  $1,803  for 
rent  in  2yrs.  Smos.  15  da,,  what  rate  of  interest  do  I  get 
for  my  money  ? 

9.  What  sum  of  money,  at  6%,  will  produce,  in  2  yrs, 
9  mos.  10  da.,  the  same  interest  that  1350  produces,  at  S%, 
in  Syrs.  lO-mos.  6  da  J 

10.  In  what  time  will  15,000  at  11%,  produce  the  same 
interest  as  $9,625  at  6^^,  in  4: yrs,  5  mos.  18  daJ 

ANNUAL     I  NTEREST. 

193.  Annual  Interest  is  simple  interest  on  the  prin- 
cipal and  also  on  each  year's  interest  from  the  time  it  falls 
due  to  the  time  of  settlement. 

This  mode  of  computation  is  legal  in  some  of  the  States  when 
notes  are  made  payable  "  with  interest  annually." 

EXAM  PLES. 

1.  What  is  the  interest  on  a  note  for  $600  at  6%,  payable 
in  3  years  tvith  interest  annually? 

Solution.— The  interest  on  $600  for  3  yrs.  is  $108 ;  the  interest 
on  $36  (the  first  year's  interest)  for  2  yrs.  is  $4.32  ;  and  the  interest 
on  $36  {the  second  year's  interest)  for  1  yr.  is  $2.16 :  hence,  the  entire 
interest  is  equal  to  $108 +  $4.32 +  $2. 16  =  $114.48.  A7is. 

2.  What  is  the  interest  on  a  note  for  $1,200  at  7^,  pay- 
able in  4  yrs.  with  annual  interest  ? 

3.  What  is  the  interest  on  a  note  for  $980  at  %%,  payable 
in  4: yrs.  with  ammal  interest? 


SIMPLE     INTEREST.  223 

NOTES. 

•    194.  A  Promissory  Note  is  a  written  promise  to  pay 

a  sum  of  money,  either  on  demand,  or  at  some  specified 

time. 

The  person  who  signs  the  note  is  called  the  Maker,  and  the  party 
that  has  legal  possession  of  it  is  called  the  Holder. 

195.  A  Negotiable  Note  is  one  that  is  payable  either 
to  order,  or  to  bearer. 

FORM     OF     A     NEGOTIABLE     NOTE. 

In  this  case  John  Doe,  the  person  named  in  the  note,  is  called  the 
Payee  ;  he  can  transfer  it  by  writing  his  name  across  the  back  ;  he 
is  then  called  an  Indorser,  and  is  obliged  to  pay  the  money  when  it 
falls  due,  if  the  maker  of  the  note,  Richard  Roe,  fails  to  do  so. 

196.  The  Face  of  a  note,  or  bill,  is  the  sum  named 
in  it.     Thus,  in  the  note  just  above  described,  the  face  is 

$375. 

PARTIAL      PAYM  ENTS. 

197.  A  Partial  Payment  is  a  payment  of  a  part  of 
the  amount  due  on  a  note,  or  other  written  obligation  to 
pay  money. 

The  date  and  the  amount  of  each  partial  payment  is  indorsed, 
that  is,  written  on  the  back  of  the  note,  or  obligation,  and  is  to  be  taken 
into  account  in  making  the  settlement. 


224  PERCENTAGE    AND     ITS     APPLICATIONS. 

METHODS    OF    SETTLEMENT. 

198.  The  following  method  of  settling  a  note,  or  other 
interest-bearing  obligation  on  which  partial  payments  have 
been  made,  has  been  sanctioned  by  the  Supreme  Court  of 
the  United  States,  and  is  now  adopted  in  New  York, 
Massachusetts,  and  many  other  States  : 

SUPREME  COURT  RULE. 

/.  Find  the  amount  of  the  given  principal  up  to 
the  time  when  the  sum  of  the  partial  payments 
is  equal  to,  or  exceeds,  the  interest  then  due ;  from 
this  result  subtract  the  sum  of  the  partial  pay- 
ments to  the  time  considered. 

II.  Take  the  remainder  for  a  new  principal  and 
proceed  as  before,  continuing  the  operation  to  the 
time  of  final  settlement. 

EXAMPLES. 

1.  On  a  note  dated  May  1, 1866,  for  f  1,200  at  6^,  were 
the  following  indorsements:  Nov.  1,  1866,  $100;  Mar.  1, 
1867,  $20;  Sept.  1,  1868,  1180:  what  was  due  on  the 
note  Nov.  1, 1869  ? 

OPERATION, 

Given  principal $1200 

Int.  to  Nov.  1,  1866  (6  mo8.) $86 

Amount $1236 

1st  payment $100 

1st  new  principal $1136 

Int.  to  Sept.  1,  1868  (22  moa)      ....     $12496 

Amount $1260.96 

Sum  of  2d  and  3d  payments $200 

Second  new  principal $1060.96 

Int.  to  Nov.  1st,  1869  (14  mos)     ....       $74.267 

Amount  due  Nov.  1st,  1869 $1135.227 


SIMPLE     INTEREST.  225 

Explanation. — We  compute  the  interest  on  $1,200  from  the 
date  of  the  note  to  the  time  of  the  first  payment,  and  because  the 
first  payment  is  greater  than  the  interest  then  due,  we  add  the 
interest  to  the  principal  and  subtract  the  first  payment,  which 
gives  $1,136  for  a  new  principal.  We  then  see,  by  inspection,  that 
the  interest  on  the  new  principal  from  Nov.  1,  1866,  to  Mar,  1, 
1867  {4:mos.),  is  greater  than  the  second  payment;  we  therefore 
compute  the  interest  on  the  new  principal  to  the  time  of  the  third 
payment,  add  it  to  the  principal,  and  from  the  result  subtract  the 
sum  of  the  2d  and  3d  payments,  which  gives  another  new  prin- 
cipal. We  then  find  the  amount  of  this  principal  up  to  the  time 
of  settlement.. 

2.  On  a  note  dated  May  1,  1866,  for  $35(>  at  6%,  were 
the  following  indorsements : 

Dec.    25,  1866,  received  150. 
Sept.     1,  1868,        "        120. 
June  13,  1869,        "      $100. 
What  was  due  April  13,  1870  ? 

3.  On  a  note  dated  July  1, 1869,  for  $700  at  7^,  were 
the  following  indorsements : 

July  1,  1870,  received  1200. 
July  1,  1871,        "        $400. 
What  is  due  July  1,  1872  ? 

4.  On  a  note  dated  May  1, 1860,  for  $2,000  at  1!%,  were 
the  following  indorsements : 

May  1,  1861,  received  $500. 
May  1,  1862,  "  $450. 
May  1,  1863,  «  $750. 
May  1,  1864,        «        $400. 

What  was  due  May  1,  1865  ? 

g  Ans,   S311.761. 


226  PERCENTAGE     AND     ITS     APPLICATIONS. 

5.  On  a  note  dated  Aug.  1, 1870,  for  $1,000  at  6%,  were 
the  following  indorsements : 

Dec.    1,  1870,  received  1310.25. 
April  1,  1871,        "        $225.50. 
Aug.  1,  1872,        "        $400.00. 
What  was  due  Jan.  1,  1873  ? 

6.  On  a  note  dated  Sept.  18, 1873,  for  17,000  at  6%,  were 
the  indorsements:  July  6,  1874,  $500;  Sept.  24,  1875, 
$1,500;  and  Dec.  6,  1875,  $1,000:  what  was  due  July  12, 
1876? 

7.  On  a  note  dated  Jan.  6,  1875,  for  $1,280  at  7^,  were 
the  indorsements:  July  18,  1875,  $175;  Dec.  12,  1875, 
$375 ;  and  July  24,  1876,  $400 :  what  was  due  July  9, 
1877  ? 

8.  On  a  note  dated  June  15, 1874,  for  $1,500  at  7^,  were 
the  indorsements:  Dec.  15,  1874,  $300;  May  30,  1875, 
$300;  and  Dec.  18,  1875,  $400:  what  was  due  July  15, 
1877  ? 

When  partial  payments  are  made  on  interest-bearing  obligations 
due  witliin  a  year,  the  balance  is  usually  adjusted  amongst  business 
men  by  the  following  rule,  called 

MERCANTILE     RULE. 

Find  the  amount  of  the  principal  from  the  date 
of  the  note  to  the  time  of  settlement ;  find  the 
amount  of  each  paym^ent  from  the  time  it  ivas 
made  to  the  time  of  settlement,  and  subtract  their 
sum  from  the  first  result. 

Note. — In  applying  this  rule,  the  times  are  reduced  to  days,  and 
the  interest  is  then  computed  by  the  rule  for  days. 


SIMPLE     INTEREST.  227 

9.  On  a  note  dated  Jan.  1,  1873,  for  $1,000  at  1%,  were 
the  following  indorsements  : 

Feb.  15,  1873,  received  $200.  .      . 

May  16,  1873,        "        $400. 

What  was  due  Aug.  14,  1873  ? 

SOLUTION. 

Amt.  of  $1000  for  225rf« $1043.75 

Amt.  of    $200  for  imda $207 

Amt.  of    $400  for    ^0  da $407 

Sum  of  amts.  of  payments $614 

Balance  due  Aug.  14,  1873       .     .     .     $429.75 

10.  On  a  note  dated  Jan.  1, 1873,  for  $800  at  6^,  are  the 
following  indorsements : 

Feb.      6,  1873,  received  $200. 
April  30,  1873,        "        $210. 
What  is  due  on  the  note  June  5,  1873  ? 

REVIEW     QUESTIONS. 

(183.)  What  is  interest?  (184.)  What  is  the  principal? 
The  rate?  The  amount?  What  is  the  legal  rate?  What  is 
usury?  How  is  time  reckoned?  (185.)  What  is  simple  inter- 
est? Compound  interest?  (186.)  Rule  for  interest  when  the 
principal,  rate,  and  time  in  years  are  given?  (187.)  When  the 
time  is  given  in  months  ?  When  in  years,  months,  and  days  ? 
(188.)  Rule  for  interest  when  the  time  is  given  in  days?  (189.) 
Rule  for  accurate  interest?  (190.)  Rule  for  finding  the  rate? 
(191.)  Rule  for  finding  the  time?  (192.)  Rule  for  finding  the 
principal?  (193.)  What  is  annual  interest?  (194.)  What  is  a 
promissory  note?  The  maker?  The  holder?  (195.)  What  is  a 
negotiable  note ?  Payee?  Indorser?  (196.)  What  is  the  face  of 
a  note  or  other  obligation?  (197.)  What  is  a  partial  payment? 
(198.)  What  is  the  Supreme  Court  rule  ?    The  Mercantile  rule  ? 


238 


PERCENTAGE     AND     ITS     APPLICATIONS. 


VII.    COMPOUND    INTEREST. 

DEFINITIONS. 

199.  Compound   Interest  is  interest  computed  on 

the  principal  and  also  on  the  accrued  interest  as  it  falls 

due. 

Interest  may  be  added  to  principal  at  the  end  of  each  year,  half 
year,  or  other  fixed  period.  Unless  otherwise  stated,  the  period  is 
supposed  to  be  a  year. 

200.  From  principles  already  explained  we  have  the 
following 

RULE     FOR     COMPOUND     INTEREST. 

I.  Find  the  amount  of  the  given  principal  for 
the  first  period  ;  then  find  the  amount  of  this  result 
for  the  second  period ;  and  so  on  to  the  end  of  the 
given  tiine ;  the  final  result  will  he  the  total  amount. 

II.  From  the  total  amount  subtract  the  given 
principal  and  the  remainder  will  be  the  compound 
interest. 

EXAMPLES. 

1.  What  is  the  compound  interest  on  $642  for  2  years 
at  6^  per  annum,  interest  being  compounded  annually  ? 


Explanation.  — Here 
the  period  is  1  year, 
and  the  amount  of  $1 
for  that  period  is  $1.06  ; 
multiplying  $642  by  1.06, 
we  find  the  amount  for 
the  first  period,  $680.52  ; 
multiplying  this  by  1.06, 
we  find  its  amount  for  the 
second  period,  $721,351. 
Subtracting  th,e  original 


$642. 
1.06 

$680.52. 
1.06 

$721.351 . 

$642 


OPERATION. 

. . .  Principal, 
. . .  1st  amount, 
. .   Total  amount, 


$79.351 . .    .  Compound  int 
principal,  we  find  $79,351  for  the  required  interest. 


COMPOUKD     IKTEREST. 


2^9 


2.  What  is  the  compound  interest  on  $918  for  3  years 
at  Q%  per  annum,  interest  being  compounded  annually  ? 

3.  What  is  the  amount  of  $650  for  4  years  at  6%  per 
annum,  interest  being  compounded  semi-annually  ? 

Explanation. — Here  the  period  is  one  half  of  a  year,  and  the 
amount  of  $1  for  each  period  is  $1.03.  Proceeding  as  before,  we 
find  the  amount. 

Note. — The  operation  of  computing  the  amount  of  any  sum  for 
a  given  time  may  be  shortened  by  the  use  of  the  following 


TABLE, 

Sliowiny  the  amount  of  $1  at  compound  interest,  for  any  number 
of  periods  from  1  to  20. 


PERIODS. 

2%. 

^fc. 

3%. 

Hfc. 

4%. 

5%. 

6%. 

11%. 

'             1 

1.0200 

1.0250 

1.0300 

1.0350 

1.0400 

1.0500 

1.0600 

1.0700 

2 

1.0404 

1.0506 

1.0609 

1.0712 

1.081G 

1.1025 

1.1236 

1.1449 

3 

1.0612 

1.0769 

1.0927 

1.1087 

1.1249 

1.1576 

1.1910 

1.2250* 

4 

1.0824 

1.1038 

1.1255 

1.1475 

1.1699 

1.2155 

1.2625 

1.3108 

5 

1.1041 

1.1314 

1.1593 

1.1877 

1.2167 

1.2763 

1.3382 

1.4026 

6 

1.1262 

1.1597 

1.1941 

1.2293 

1.2653 

1.3401 

1.4185 

1.5007 

7 

1.1487 

1.1887 

1.2299 

1.2723 

1.3159 

1.4071 

1.5036 

1.6058' 

8 

1.1717 

1.2184 

1.2668 

1.3168 

1.3686 

1.4775 

1.5938 

1.7182 

9 

1.1951 

1.2489 

1.3048 

1.3629 

1.4233 

1.5513 

1  6895 

1.8385 

10 

1.2190 

1.2801 

1.3439 

1.4106 

1.4802 

1.6289 

1.7908 

1.9672, 

11 

1.2434 

1.3121 

1.3842 

1.4600 

1.5395 

1.7103 

1.8983 

2.1049! 

12 

1.2682 

1.3449 

1.4258 

1.5111 

1.6010 

1.7959 

2.0122  2.2522 

13 

1.2936 

1.3785 

1.4685 

1.5640 

1.6651 

1.8856 

2.132912.4098 

14 

1.3195 

1.4130 

1.5126 

1.6187 

1.73171.9799 

2.26092.5785 

15 

1.3459 

1.4483 

1.5580 

1.6753 

1.8009|2.0789 

2.3966  2.7590 

16 

1.3728 

1.4845 

1.6047 

1.7340 

1.8730j2.1829 

2.5404  2.9522 

17 

1.40021.5216 

1.6528 

1.7947 

1.9479  2.2920 

2.6928  3.1588 

18 

1.42821.55971.7024 

1.8575 

2 .  0258  2 .  4066  2 .  8543  3 .  3799 

19 

1.4568 

1.5987 

1.7535 

1.9225 

2.1068:2.5270 

2.0256,3.6165! 

20 

1.4859 

1.6386 

1.8061 

1.9898 

2.19112.6533 

2. 2071 j 

3.8697! 

J 

230  PEHCEiTTAGE    AND     ITS    APPLICATIONS. 

4.  What  is  the  amount  of  $820  for  Qyrs.  at  4^  per 

annum,  interest  being  compounded  semi-annually? 

Explanation. — In  this  case  there  are  12  periods  and  the  rate  for 
each  period  is  2  % .  From  the  table,  we  find  that  |1  at  the  given 
rate  amounts  to  $1.3682  in  that  time,  and  because  $820  amounts  to 
820  times  as  much  as  $1,  we  have,  $1 .2682  x  820  =  $1,039,924.  Ans. 

5.  What  is  the  amount  of  $900  for  9  years  at  7^  per 
annum,  interest  being  compounded  semi-annually? 

6.  What  is  the  compound  interest  on  $1,850  for  3  yrs, 
at  8^,  interest  being  compounded  quarterly  ? 

7.  Find  the  amount  of  $800  for  14  years  at  7^  per 
annum,  interest  being  compounded  annually. 

Ans.  $2,062.80. 

Note. — If  the  last  period  is  fractional,  compute  the  amount  to 
the  end  of  the  next  preceding  period,  and  then  find  the  amount  of 
that  result  for  the  fractional  period. 

8.  What  is  the  amount  of  $500  for  3  years  2  months,  at 

Q%  per  annum,  interest  being  compounded  annually  ? 

Explanation.— The  amount  for  3  years  is  $1,191  x  500  or  $595.50, 
and  this  in  two  months  amounts  to  $601,455.  Ans. 


9.  What  is  the  amount  of  $1,200  for  4ryrs,  Smos.  at  7%, 
interest  being  compounded  annually  ? 

Ans.  $1,646,365. 

10.  What  is  the  amount  of  $1,350  for  6 yrs.  4:mos.  at 
6%,  interest  being  compounded  semi-annually  ? 

A71S.  $1,850.55. 
Note. — In  using  the  table  we  retain  the  4  decimal  places  given. 

REVIE^A^      QUESTIONS. 

(199.)  What  is  compound  interest  ?  How  often  may  interest  be 
added  to  j^rincipal  ?  (200.)  Wliat  is  the  rule  for  compound 
interest  V    Explain  the  use  of  the  table. 


DISCOUNT.  231 


VIII.      DISCOUNT. 

301.  Discount  is  a  percentage  deducted  from  the 
face  of  a  bill,  debt,  or  note. 

COMMERCIAL    DISCOUNT. 

203.  Commercial  Discount  is  a  percentage  de- 
ducted from  the  face  of  a  bill  of  merchandise. 

The  face  of  the  bill  is  the  Base  and  the  difference  be- 
tween this  and  the  discount  is  called  the  Net  Proceeds. 

From  definitions  and  preceding  principles  we  have  the 
following 

RULE. 

/.  Multiply  the  face  of  the  hill  by  the  rate  per 
c&nt.  and  the  product  will  he  the  discount. 

II.  Subtract  the  discount  from  the  face  of  the 
bill  and  the  difference  will  he  the  net  proceeds. 

Note.— The  net  proceeds  may  be  found  by  multiplying  the  face 
of  the  bill  by  1  minus  the  rate  per  cent. 

EXAM  PLES. 

1.  What  is  the  discount  on  a  bill  of  $350  at  h%,  and 
what  is  the  net  proceeds  ? 

Ans.  Dis.  =  117.50 ;  net  proceeds  =  1332.50. 

2.  Sold  a  bill  of  merchandise  amounting  to  11,173, 
deducting  10^  for  cash ;  what  was  the  net  proceeds  ? 

3.  Flour  is  sold  on  credit  at  $12.50  per  barrel ;  what  is 
the  cash  price,  the  discount  being  15^  ? 

4.  Find  the  discount  on  a  bill  of  goods  whose  face  is 
$1,200,  at  the  rate  of  ^%, 


232  PERCENTAGE     AND     ITS     APPLICATIONS. 

5.  Sold  a  lot  of  goods  amounting  to  $918,  deducting 
124^^  for  cash ;  what  was  the  net  proceeds  ? 

6.  What  is  the  net  proceeds  of  56  tubs  of  butter,  each 
weighing  42J  lbs.  at  22  cts.  a  pound,  b%  off  for  cash  ? 

7.  Sold  50  hUs.  flour  at  17.50  per  barrel,  deducting  7^^ 
for  cash ;  what  was  the  net  proceeds  ? 

8.  Sold  coal  at  15.50  per  ton,  10%  off  for  cash ;  what  was 
the  cash  price  ? 

9.  Sold  500  5w.  of  oats  at  Q,^cts.  a  bushel,  b%  off  for 
cash  ;  what  was  the  cash  price  ? 

PRESENT  VALUE  AND  TRUE  DISCOUNT. 

203.  The   Present   Value   of  a  debt  payable  at  a 

future  time  is  a  sum  which,  being  placed  at  interest,  will 

give  an  amount  equal  to  the  debt  when  it  falls  due.    Thus, 

$100  is  the  present  value  of  $107  due  1  year  hence,  interest 

being  reckoned  at  1%. 

If  the  debt  does  not  bear  interest,  its  amount  is  the  same  as  its 
face ;  if  it  bears  interest,  its  amount  is  equal  to  its  /crce  together 
with  the  interest  up  to  the  time  it  is  due. 

The  True  Discount  is  the  difference  between  the 
amount  of  the  debt  and  its  present  value. 

The  method  of  finding  the  present  value  of  a  debt  due 
at  a  future  time  is  the  same  as  finding  the  principal,  when 
the  amount,  the  rate,  and  the  time  are  given ;  hence,  the 
following 

RULE. 

I.  Divide  the  amount  of  the  debt  by  1  plus  the 
product  of  the  rate  by  the  time  in  years;  the 
quotient  will  be  the  present  value. 


DiscouiTT.  233 

II.  Subtract  the  present  value  from  the  amount 
of  the  debt ;  the  remainder  will  be  the  true  discount. 

EX  A  M  PLES. 

1.  What  is  the  present  value  of  $1,500,  due  1  yr.  4  mos. 
hence,  money  being  worth  6^  per  annum  ? 

Solution.— The  rate,  .06,  multiplied  by  the  time  in  years,  1^, 
equals  .08  ;  hence,  $1,500  h-  1.08  =  $1,388,889.    Arts, 

2.  What  is  the  true  discount  on  $1,200,  due  2  years 
hence,  money  being  worth  11%  ? 

Ans.  $1,200  —  $1,200  -^  1.14  =  $147.37, 

3.  What  is  the  present  value  of  a  debt  of  $1,760,  due 

3  yrs.  6  mos,  hence,  at  the  rate  of  6^  per  annum  ? 

4.  Find  the  true  discount  on  a  debt  of  $1,141.25,  due 

7  mos.  15  days  hence,  at  6^  per  annum. 

5.  What  is  the  true  discount  on  $730,  due  in  2  years, 
at  b%  per  annum  ? 

6.  Find  the  present  value  of  a  debt  of  $986,  due  2  yrs. 

8  mos.  hence,  at  Q%  per  annum. 

7.  What  is  the  true  discount  in  the  last  example? 

8.  What  is  the  present  value  of  $1,200,  due  in  lyr, 

4  mos.,  at  7^^  per  annum  ? 

9.  A  debt  of  $1,400  is  due  in  9  mos. ;   what  is  the  true 
discount,  interest  being  computed  at  the  rate  of  Q%  ? 

10.  Find  the  present  value  of  a  note  for  $750,  due  in 
1  yr.  8  mos.  12  da.,  at  Q%. 

11.  What  is  the  present  value  of  a  note  for  $1,300,  due 
in  2  yrs.  8  mos.  at  7^  ? 

12.  What  is  the  present  value  of  $10,000,  due  in  4  mos. 
18  da.,  at  ^1%  ? 


234  PERCENTAGE     AND     ITS     APPLICATIONS. 

13.  What  is  the  present  yalue  of  $1,828.75,  due  in 
1  year,  and  bearing  4^%  interest  ? 

14.  What  is  the  present  value  of  a  note  for  $4,800,  due 
4  yrs.  hence,  at  5%  interest  ? 

15.  A.  owes  B.  $3,456,  payable  Oct.  27,  1877;  what 
ought  A.  to  pay  Aug.  24,  1877,  interest  being  Q%  per 
annum  ? 

BANKS    AND    BANK    DISCOUNT. 

304.  A  Bank  is  an  incorporated  institution  author- 
ized by  law  to  deal  in  money. 

Some  of  tlie  operations  of  banking  are,  receiving  money  for  safe 
keeping,  discounting  notes  and  other  evidences  of  indebtedness, 
issuing  bills  to  circulate  as  money,  buying  and  selling  bills  of 
exchange,  gold  and  silver  bullion,  coin,  and  the  like.  Most  banks 
are  engaged  in  only  a  part  of  these  operations. 

205,  Bank   Discount  is  a  percentage  charged  for 

advancing  money  on  a  note  or  other  obligation  payable  at 

a  future  time  ;  it  is  simply  interest  in  advance  on  what  the 

note  or  obligation  will  yieeld  when  it  becomes  legally  due. 

This  operation  of  advancing  money  on  notes  or  obligations  not 
yet  due  is  called  Discounting. 

206.  A  note  is  said  to  Mature  when  it  becomes 
legally  due,  which  is  three  days  after  it  is  nominally  due. 
The  three  days  that  elapse  after  a  note  is  nomi7ially  due 
before  it  is  legally  due  are  called  Days  of  Grace. 

The  time  at  which  a  note  falls  due  may  be  denoted  by  a  double 

date.    Thus,  the  expression,  "  Due  July  /(a/'  written  at  the  foot  of 

a  note  shows  that  it  is  nominally  due  on  the  7th  of  July,  and  that  it 
is  legally  due  on  the  10th  of  July. 

Note. — If  the  last  day  of  grace  falls  on  Sunday,  or  on  a  legal 
holiday,  the  note  is  legally  due  in  some  of  the  States  on  the  preced- 
ing day,  and  in  others  on  the  foUomng  day. 


DISCOUNT.  235 

METHOD    OF    DISCOUNTING    A    NOTE. 

207.  The  ordinary  method  of  discounting  a  note  is 
illustrated  in  the  following  example : 

Form  of  Note. 

$600. 

New  Yoek,  July  7,  1877. 

Sixty  days  after  date  I  promise  to  pay  to  the  order 

of  Lucius   Slocum   six  hundred   dollars  at   the  Fourth 

National  Bank.    Value  received. 

KoBEET  Bull. 
Due  Sept.  Vg,  1877. 

Explanation. — This  note  is  supposed  to  be  discounted  on  the 
day  of  its  date.  The  holder,  Lucius  Slocum,  endorses  it  by  writing 
his  name  across  its  back,  and  delivers  it  to  the  proper  bank  officer. 
Interest  is  then  computed  on  the  face  of  the  note,  $600,  for  63  days, 
the  days  of  grace  being  included,  and  at  the  legal  rate,  which  in  New 
York  is  7%.  This  sum,  S7.35,  is  the  discount  ;  subtracting  the  dis- 
count from  $600,  we  have  $592.65,  which  is  called  the  Proceeds,  and 
this  amount  is  paid  over  to  Lucius  Slocum.  If  the  note  is  not  paid 
before  the  close  of  the  last  day  of  grace,  Sept,  8,  a  written  notice, 
called  a  Protest,  is  sent  to  Lucius  Slocum,  and  he  then  becomes 
liable  for  its  payment. 

EXAMPLE. 

1.  A  note  for  $1,400,  payable  60  days  after  date,  is  dis- 
counted at  the  rate  of  7^;  what  is  the  proceeds  ? 

Explanation.— The  interest  of  $1,400  for  63  days  at  7%  is 
$17.15  ;  this  is  the  discount :  subtracting  it  from  $1,400,  we  have 
$1,382.85,  which  is  the  proceeds. 

Note. — If  a  note  is  not  discounted  on  the  day  of  its  date,  the  dis- 
count is  reckoned  from  the  time  of  discount  to  the  time  of  maturity. 

EXAMPLE. 

2.  A  note  for  $1,200,  dated  Sept.  5,  1877,  and  payable 
90  days  after  date  at  7^,  is  discounted  Oct.  2,  1877;  what 
is  the  proceeds  ?  Ans.  $1,184.60. 


23  G  PERCEl^TAGE    AlTD    ITS    APPLICATIONS. 

Explanation.— Here  the  note  is  due  Dec.  ^/^,  and  from  Oct. 
2  to  Dec.  7  is  66  days  The  interest  on  $1,200  for  66  days  at  7%  is 
$15.40  ;  hence,  the  proceeds  equals  $1,200  —  $15.40,  or,  $1,184.60. 

From  what  precedes  we  have  the  following 

RULE. 

/.  Compute  interest  on  the  face  of  the  note,  from 
the  time  of  discount  to  the  time  of  jnaturity ;  this 
will  be  the  discount. 

II.  Subtract  the  discount  from  the  face  of  the 
note ;  the  result  will  be  the  proceeds. 

Note. — If  an  interest-bearing  note  is  discounted,  the  amount  of 
the  note  at  maturity  is  made  the  base  on  which  discount  is  reck- 
oned. 

E  X  AM  PLE  S. 

3.  What  is  the  bank  discount  on  a  note  for  $670,  pay- 
able 60  days  after  date  at  Q%  ? 

4.  What  is  the  bank  discount  on  a  note  for  $350,  pay- 
able 00  days  after  date  at  7%  ? 

5.  What  is  the  proceeds  of  a  note  of  $1,000,  payable  at 
bank,  60  days  after  date,  at  Q%  ? 

6.  A.  has  a  note  against  B.  for  $1,728,  payable  90  days 
after  date,  which  he  gets  discounted  at  the  rate  of  7% ; 
what  does  he  receive  ? 

7.  A  note  for  $1,620,  dated  July  7,  1877,  and  payable 
90  days  after  date,  is  discounted  July  25,  1877,  at  S%; 
what  is  the  proceeds  ? 

Note. — In  what  precedeo,  interest  has  been  computed  on  the  sup- 
position that  360  days  make  a  >oar.  If  accurate  interest  is  required, 
as  it  is  in  many  of  the  States,  it  may  be  found  by  diminishing  the 
interest,  computed  as  above,  by  its  7V  }>a''t,  or,  better  still,  by  means 
of  tables  constructed  for  the  purpose. 


DISCOUNT.  237 

208.  To  find  the  Face  of  a  Note  payable  at 
a  Future  Time,  whose  Proceeds  shall  equal  a 
Given  Sum. 

To  find  the  face  of  a  note,  payable  90  days  after  date, 
that  will  yield  $500,  the  rate  being  6%. 

Explanation.— The  proceeds  of  $1  for  93  days  at  6%  is  $.9845. 
But  the  entire  proceeds  must  be  $500  ;  hence,  the  face  of  the  note 
must  be  as  many  dollars  as  .9845  is  contained  in  500,  that  is,  it  must 
be  1500  -5-  .9845,  or  $507,872. 

In  like  manner  all  similar  cases  may  be  treated ;  hence,  the  fol» 
lowing 

RULE. 

Divide  the  given  sum  by  the  proceeds  of  $1  for 

the  given  time  and  rate;  the  quotient  will  he  the 

numher  of  dollars  required. 

EXAMPLES. 

1.  Find  the  face  of  a  note  payable  in  90  days  at  1%,  so 
that  the  proceeds  shall  be  $2,050.  Ans.  12,087.747. 

2.  Find  the  face  of  a  note  payable  60  days  after  date  at 
Q%  that  will  yield  $500. 

3.  Find  the  face  of  a  note  payable  90  days  after  date  at 
8^  that  will  yield  $750. 

4.  Find  the  face  of  a  note  payable  30  days  after  date  at 
10%  that  will  yield  $1,000. 

REVIEW      QUESTIONS. 

(201.)  What  is  discount?  (202.)  What  is  commercial  dis- 
count ?  Rulg  ?  (203.)  What  is  the  present  value  of  a  debt  ? 
What  is  true  discount  ?  Rule  for  present  value  and  true  discount  ? 
(204.)  What  is  a  bank  ?  What  operations  may  be  carried  on  by  a 
bank  ?  (205.)  What  is  bank  discount  ?  What  is  discounting  ? 
(206.)  When  does  a  note  mature  ?  What  are  days  of  grace  ? 
(207.)  Explain  the  process  of  discounting  a  note.  When  and  how 
is  a  note  protested  ?  Give  rule  for  finding  the  bank  discount  and 
proceeds  of  a  note.  (208.)  What  is  the  rule  for  finding  the  face 
of  a  note  that  will  yield  a  given  sum  ? 


238  PERCEl^TAGE     AND     ITS     APPLICATIONS. 

IX.      STOCKS    AND    BONDS. 

DEFINITIONS. 

209.  A  Corporation  is  an  association  of  persons 
authorized,  under  certain  restrictions,  to  transact  business 
as  an  individual. 

210.  The  Capital  Stock  is  the  money  used  in  car- 
rying on  the  business  of  a  corporation.  It  is  divided  into 
equal  parts  called  Shares. 

The  owners  of  the  shares  are  called  Stockholders. 

211.  A  Certificate  of  Stock  is  a  certificate  signed 
by  proper  authority,  showing  that  the  party  therein  named 
owns  a  certain  number  of  shares  of  the  capital  stock. 

212.  The  Par  Value  of  a  stock  is  the  value  named 
on  the  face  of  the  certificate.  This  is  usually  $100  per 
share,  but  it  may  be  either  more  or  less. 

213.  The  Market  Value  of  a  stock  is  the  price  it 
will  bring  in  open  market. 

If  the  market  value  is  greater  than  the  par  value,  the 
stock  is  said  to  be  at  a  Premium,  or  Above  Par ;  if 
the  market  value  is  less  than  the  par  value,  it  is  said  to  be 
at  a  Discount,  or  Below  Par. 

214.  Dividends  are  percentages  on  the  capital  stock 
paid  to  stochholders  as  profits  on  the  business.  Assess- 
ments are  percentages  that  stockliolders  are  called  on  to 
pay  to  meet  losses  and  extraordinary  expenses. 

215.  A  Bond  is  a  properly  authenticated  obligation 
to  pay  a  sum  of  money  at  or  before  a  certain  time,  with 
interest  at  fixed  periods. 


STOCKS     AND     BOI^DS.  239 

There  are  two  classes  of  bonds :  1\  Bonds  for  whose  payment 
the  public  faith  is  pledged,  as  National  and  State  bonds  ;  and  2°. 
Bonds  for  whose  payment  the  property  of  some  incorporated  com- 
pany is  pledged,  as  Railroad  bonds. 

216.  The  face  of  a  bond  is  usually  $1,000,  but  it  may 

be  any  sura,  either  greater  or  less. 

Stocks  and  bonds  are  bought  and  sold  in  open  market.  The 
reported  price,  or  Quotation,  is  the  number  of  per  cent,  that  the 
selling  price  is  of  the  par  value.  Thus,  if  a  stock  or  bond  is  quoted 
at  87|,  its  selling  price  is  87|%  of  its  par  value. 

UNITED    STATES    BONDS. 

217.  The  interest-bearing  debt  of  the  United  States  is 
represented  by  bonds  issued  at  separate  times,  payable  at 
different  dates,  and  bearing  various  rates  of  interest. 
Some  bonds  carry  certificates  of  interest,  called  Coupons, 
and  pass  from  hand  to  hand  like  bank  bills ;  others  are 
Registered  on  the  books  of  the  treasury,  and  can  only 
be  transfered  by  a  change  of  record. 

United  States  Bonds  are  designated  by  giving  their  date  of  pay- 
ment, and,  if  necessary,  their  interest.  Thus,  the  term  **  TJ.  S.  6's, 
81  R."  stands  for  "  Registered  6%  Bonds  of  the  United  States,  pay- 
able in  1881;"  the  term  "U.  S.  5-20's,  65  C."  stands  for  *' United 
States  Coupon  Bonds,  payable  at  the  option  of  the  government  at 
any  time  between  5  and  20  years  after  their  date,"  that  is,  at  any 
time  between  1870  and  1885  ;  the  tenn  "  U.  S.  10-40's  "  stands  for 
"  United  States  Bonds,  payable  at  any  time  between  10  and  40  years 
after  their  date,"  which  is  1864.  All  the  5-20's  bear  6%  interest, 
and  all  the  10-40's  bear  5  %  interest,  payable  in  gold. 

218.  All  problems  relating  to  the  buying  and  selling 

of  stock  and  bonds  are  solved  by  the  rules  for  percentage 

and  interest. 

Note. — In  what  follows  the  par  value  of  each  share  of  stock  is 
supposed  to  be  $100,  and  the  par  value  of  each  bond  $1000  ;  the 
brokerage,  which  is  paid  by  the  party  for  whom  the  purchase  or  sale 


240  PERCENTAGE     AKD     ITS     APPLICATIONS. 

is  made,  is  supposed  tohe^fo,  and  is  always  computed  on  the  par 
value  of  the  stock  or  bond. 

EXAM  PLES. 

1°.  Applications  of  the  Rules  for  Percentage 
(Arts.  165-168). 

1.  What  is  the  market  value  of  120  shares  of  bank  stock 
@  53}^  ?  Ans.  $12,000  x  .5375  =  16,450. 

2.  What  is  the  cost,  including  brokerage,  of  44  shares  of 
bank  stock  @  129^^  ?     Ans.  14,400  x  1.2975  =  15,709. 

3.  Sold  15,000  in  gold  @  lVi\%]  what  was  the  net  pro- 
ceeds after  deducting  brokerage.  A7is.  $5,600. 

4.  A.  sold  160  shares  railroad  stock  @  92|^,  and  pur- 
chased with  the  proceeds  bank  stock  @  73|-%',  paying 
brokerage  on  both  sa,le  and  purchase ;  how  many  shares 
did  he  receive  ?  Ans,  160  x  92^  -^  74  =  200. 

5.  What  is  the  premium  on  88  shares  of  bank  stock 
@  114^?  Ans.  18,800  x  .14  =  $1,232. 

6.  What  is  the  discount  on  190  shares  of  railroad  stock 
which  is  sold  @  89^  ? 

7.  A.  sold  93  bonds,  U.  S.  5-20's,  R.  @  113^^,  paying 
brokerage  on  the  sale ;  what  did  he  receive  ? 

8.  A  speculator  buys  225  shares  of  Erie  stock  @30J^ 
and  sells  it  again  @  31^;^,  paying  brokerage  on  both  trans- 
actions ;  what  does  he  gain  ? 

9.  If  170  railroad  bonds  cost  (including  brokerage) 
$192,100,  what  is  their  market  rate  ? 

10.  What  is  the  cost,  including  brokerage,  of  $9,000 
U.  S.  6V81  C.  @112i? 

11.  Sold  $42,000  U.  S.  5-20's  C.  for  $46,830  net,  after 
paying  brokerage  ;  what  was  the  market  price  ? 


STOCKS     AND     BONDS.  241 

12.  How  much  gold  @  lllf  can  be  bought  for  $8,930  in 
currency,  brokerage  not  considered  ? 

13.  A  broker  sells  30  shares  of  bank  stock  @  96^%,  and 
120  shares  of  railroad  stock  at  105^,  retaining  the  broker- 
age ;  how  much  does  he  pay  to  his  principal  ? 

Ans.  $15,457.50. 
2°.  AppUcatmis  of  the  rules  for  interest. 

14.  If  I  buy  a  6%  stock  @  90^,  what  rate  of  interest  do 
I  receive  on  the  investment  ? 

Explanation. — Here  the  principal,  $90,  yields  an  interest  of  $6 
per  annum  ;  hence,  we  have, 

$6-f-$90  =  6|%.  Ans. 

15.  The  market  rate  of  a  6%  stock  is  85^% ;  if  the  pur- 
chaser pays  brokerage,  what  rate  of  interest  does  he  receive 
on  his  investment  ? 

16.  An  8%  stock  sells  @  112^ ;  what  rate  of  interest 
does  it  yield  to  the  purchaser  ? 

17.  At  what  rate  must  an  8%  stock  be  purchased  to 
yield  the  purchaser  7^  interest  ? 

In  this  case  the  price  of  one  share  is  a  principal  which,  at  the 
rate  of  7%,  yields  $8.00  in  one  year;  hence,  (Prin.  5°,  Art.  164), 
we  have, 

$8.00  -^  $.07  =  114f  %.  Ans. 

18.  I  wish  to  purchase  a  6%  stock  on  such  terms  that  it 
will  give  me  7%  on  my  investment ;  how  much  can  I  pay 
for  the  stock,  including  brokerage  ? 

19.  If  I  buy  U.  S.  6's,  '81,  at  112|,  including  brokerage, 
and  sell  the  gold  interest  at  107|-^  for  currency,  what  rate 
of  interest  do  I  get  on  my  investment  ? 

Ans.  $6  X  1.071  -r-  112^  =  5.75^,  nearlt/. 


24:2  PERCENTAGE    A^D    ITS    APPLICATIONS. 

20.  At  what  rate  must  I  buy  a  6%  stock  to  get  the  same 
rate  of  interest  as  from  a  6%  stock  at  116%  ?     Ans.  90%. 

REVIEW     QUESTIONS. 

(209.)  What  is  a  corporation?  (210.)  What  is  the  capital 
stock?  Shares?  Stockholders?  (211.)  What  is  a  certificate  of 
stock?  (212.)  What  is  par  value  of  a  stock?  (213.)  What  is 
the  market  value  ?  When  is  a  stock  at  a  premium  ?  At  a  discount  ? 
(214.)  What  are  dividends?  Assessments?  (215.)  What  is  a 
bond  ?  Explain  the  two  classes.  (2 16.)  How  are  bonds  and  stocks 
quoted?  (217.)  Give  an  account  of  U.  S.  bonds.  What  are  cou- 
pons and  coupon  bonds?  Registered  bonds?  (218.)  By  what 
rules  do  we  solve  problems  in  stock  operations  ? 


X.     EXCHANGE. 

DEFINITIONS. 

219.  Exchange  is  a  method  of  making  payments  at 
distant  places  by  means  of  drafts,  or  bills  of  exchange. 

The  theory  of  settling  accounts  by  exchange  may  be  illustrated 
by  the  following  simple  case :  A  flour  merchant  of  Chicago  forwards 
$5,000  worth  of  flour  to  a  shipper  in  New  York,  and  at  the  same 
time  a  dry  goods  merchant  of  Chicago  buys  $5,000  worth  of  mer- 
chandise from  a  New  York  importer.  The  flour  merchant  draws 
his  draft  for  $5,000  on  the  shipper  and  sells  it  to  the  dry  goods  mer- 
chant ;  the  latter  forwards  it  to  the  importer,  who  presents  it  to  the 
shipper  and  receives  the  money  called  for.  In  this  way  the  debts  in 
both  cities  are  liquidated  without  the  necessity  of  sending  any  money 
from  either. 

Dealers  in  exchange  usually  buy  drafts  on  distant  places  and 
send  them  forward  as  a  basis  of  credit ;  they  then  sell  their  own 
drafts  drawn  against  this  credit  in  sums  to  suit  their  customers. 

220.  A  Draft  or  Bill  of  Exchange  is  a  written 
order  from  one  party  to  another  to  pay  to  a  third  party  a 
certain  sum  of  money  at  a  specified  time. 


EXCHANGE.  243 

A  Sight  Draft  is  one  that  is  payable  on  presentation ; 

a  Time  Draft  is  one  that  is  payable  at  a  specified  time, 

or  at  a  certain  time  after  presentation. 

Note. — In  the  latter  case  three  days  grace  are  allowed,  but  not  in 
the  former. 

231.  The  Drawer  or  Maker  is  the  party  that  draws 

the  bill ;   the  Drawee  is  the  one  on  whom  it  is  drawn ; 

and  the  Payee  is  the  party  to  whom  the  money  is  to  be 

paid. 

A  party  that  buys  a  bill  of  exchange  is  called  a  Buyer,  or  Re- 
mitter ;  if  the  payee,  or  any  other  party,  writes  his  name  across  the 
back  of  the  bill,  he  is  called  an  Indorser ;  and  the  party  that  has 
legal  possession  of  the  bill  at  the  time  of  payment  is  called  the 
Holder. 

223.  An  Acceptance  is  an  agreement  on  the  part  of 
the  draicee  to  pay  the  draft  at  maturity.  If  he  agrees  to 
pay  it,  he  writes  the  word  Accepted  across  its  face  and 
signs  his  name  ;  he  is  then  called  an  Acceptor. 

223.  An  Inland  or  Domestic  Bill  is  one  in  which 
both  drawer  and  drawee  reside  in  the  same  country. 

A  Foreign  Bill  is  one  in  which  the  drawer  and  drawee 
reside  in  different  countries. 

224.  The  Par  of  Exchange  between  two  places  is 

the  relative  value  of  the  principal  units  of  currency  of  the 

two  places. 

Thus,  £1  Sterling  is  equal  in  value  to  $4.8665,  and  this  is  the 
par  of  exchange  between  London  and  New  York  ;  the  Franc  is 
equal  to  $0,193,  and  this  is  the  par  of  exchange  between  Paris  and 
New  York ;  the  Mark  is  equal  to  $0,238,  and  this  is  the  par  of 
exchange  between  Berlin  and  New  York. 

Note.— If  $4.8665  in  New  York  will  just  purchase  a  bill  of  £1  on 
London,  exchange  on  London  is  at  par  ;  if  it  will  buy  a  larger  bill, 
exchange  is  against  London  and  in  favor  of  New  York  ;  if  it  requires 


244  PERCENTAGE     AND     ITS     APPLICATIONS. 

more  than  $48665  to  buy  a  bill  of  £1,  exchange  is  in  favor  of  Lon- 
don and  against  New  York. 

225.  The  Course  of  Exchange  is  the  variation  in 

price  of  bills  of  exchange. 

These  variations  are  shown  in  the  daily  quotations  published  in 
the  papers.  Thus,  on  the  10th  of  August  sterling  exchange  was 
quoted  at  $4.86,  sight;  that  is,  a  sight  bill  on  London  was  worth 
$4.86  for  each  pound  sterling  of  its  face. 

INLAND    OR    DOMESTIC    EXCHANGE. 

226.  Inland  or  Domestic  Exchange  is  the  method 
of  making  payments  at  distant  places  in  the  same  country 
by  means  of  drafts  or  inland  hills  of  exchange. 

Form  of  a  Draft. 

1500. 

Troy,  N.  T.,  Aug.  10,  1877. 

At  sight,  pay  to  the  order  of  W^illiam  Whitman  Five 

Hundred  Dollars,  value  received,  and  charge  the  same  to 

the  account  of 

John  S.  Wesley. 

To  the  Fourth  National  Bank, 
New  York  City. 

Note.*— To  change  the  above  to  a  time  draft,  say  for  10  days,  the 
words  "  at  sight"  are  replaced  by  "  at  ten  days  sight." 

227.  All  problems  relating  to  inland  exchange  can  be 

solved  by  the  rules  for  percentage  and  bank  discount. 

In  applying  the  rules  for  percentage,  the  Face  of  the  Bill  is  the 
base,  the  Premium  or  Discount  is  the  raie  per  cent.,  and  the  Cost 
of  the  Bill  is  the  amount. 

EXAMPLES. 

1.  What  is  the  cost  of  a  sight  draft  when  exchange  is 
1^%  premium  ? 

Solution.— $500  +  1^%  of  $500  =  $507.50.    Am. 


EXCHAI^GE.  245 

2.  What  is  the  cost  of  a  sight  draft  on  New  York  for 
11,250,  exchange  being  1^%  discount  ?    Ans.  $1,234.37|-. 

3.  What  is  the  market  price  of  a  sight  draft  on  New 
York  for  $890,  exchange  being  worth  101 J^  ? 

Explanation. — Here  the  rate  of  premium  is  l|^%,  and  1  plus  the 
rate,  that  is,  the  amount  per  cent,  is  1.0125  ;  hence,  the  draft  is 
worth  $890  x  1.0125  =  $901. 12|.     Ans. 

4.  Find  the  market  value  of  a  sight  draft  on  New  York 
for  11,800,  exchange  being  99^.  A71S.  11,782. 

Note. — Both  exchange  and  bank  discount  are  computed  on  the 
face  of  a  time  draft.  They  may  be  computed  separately,  or  together, 
as  is  more  convenient. 

5.  Find  the  cost  of  a  draft  on  New  York  for  11,400, 
payable  60  days  after  sight,  exchange  being  worth  102-J-^, 
and  interest  being  reckoned  at  '7%. 

OPERATION. 

Amount  of  $1400  at  2^%  premium     .     .     .  $1435.00 
Interest  on  $1400  for  63  days  at  7%    .     .     .         17.15 

Difference $1417.85  Ans. 

6.  What  is  the  cost  of  a  draft  on  New  York  for  12,400, 
payable  90  days  after  sight,  interest  being  10^  and  ex- 
change 103^  ?  Ans.  12,410. 

7.  Find  the  value  of  a  draft  on  New  York  for  $1,650, 
payable  60  days  after  sight,  exchange  being  98|^  and 
interest  6%.  Ans.  $1,650  x  .9745  =  $1,607,925. 

8.  If  exchange  is  101^^,  how  large  a  sight  draft  can  be 
bought  for  $7,900  ?    A71S.  $7,900  -^  I.OIJ  =  $7,783,251. 

9.  What  is  the  face  of  a  sight  bill  that  can  be  bought 
for  $5,000,  when  exchange  is  98^^  ? 

Solution.— $5,000  -^  .985  --=  $5,076.14^.    Ans. 


246  PERCENTAGE     AND     ITS     APPLICATIONS. 

10.  What  is  the  face  of  a  draft  at  60  days  sight  which 
costs  11,000,  exchange  being  103^  and  interest  6%  ? 

Explanation.  — The  interest  on  $1  for  63  days  is  $0.0105,  and 
this  taken  from  $1.03  gives  $1.0195  as  the  cost  of  $1  of  exchange. 
Hence,  the  face  of  the  bill  is  $1,000  -^  1.0195  -  $980.87^^.     Arts. 

11.  What  is  the  face  of  a  draft  at  30  days,  which  costs 
$2,000,  exchange  being  102%  and  interest  6%  ? 

FOREIGN    EXCHANGE. 

228.  Foreign  Exchange  is  the  method  of  making 
payments  in  foreign  places  by  means  of  bills  of  exchange. 

Form  of  Set  of  Exchange. 

^^^^'  New  York,  Aug.  11,  1877. 

At  sight  of  this  first  of  exchange  {second  and  third 

of  same   date  and   tenor  unpaid),  pay  to  the  order  of 

Salmon  Stoddard  Five  Hundred  Pounds  Sterling,  value 

received,  and  charge  the  same  to 

Johnson  Lummis. 
To,  Copely  &  Brothers,  London. 

Note. — Three  copies  constitute  a  set,  each  of  which  is  forwarded 
by  a  different  mail.  The  other  copies  differ  from  the  one  above 
given,  in  no  respect,  except  that  in  one  the  words  first  and  second 
change  places,  and  in  the  other  the  words  first  and  third  change 


229.  Exchange  betweeen  the  United  States  and  for- 
eign countries  is  computed  by  the  method  of  equivalents. 
The  equivalents  are  made  known,  both  for  sight  and  time 
bills,  by  the  current  or  daily  quotations.  But  as  these 
equivalents  are  in  gold,  they  must  be  converted  into  cur- 
rency by  the  ordinary  rules  for  percentage. 


EXCHAN^GE.  247 

EXAM  PLES. 

1.  What  is  the  cost  of  a  sight  bill  for  £87,  when  £1  is 

worth  $4.82  in  gold,  gold  being  worth  106^  currency  ? 

Explanation. — Because  £1  is  worth  |4.82  in  gold,  £87  is  worth 
$4.83  X  87  =  $419.34,  and  this  is  converted  into  currency  by  multi- 
plying it  by  1.06  ;  hence,  $419.34  x  1.06  =  $44450.    Ans. 

2.  What  is  the  cost  of  a  sight  bill  on  London  for  £750, 
when  £1  is  worth  $4.85,  gold  being  quoted  at  108^  ? 

3.  What  is  the  cost  of  a  sight  bill  on  London  for  £300, 
£1  being  worth  $4.86,  gold  107^  currency  ? 

4.  Find  the  cost  of  a  60  day  bill  on  London  for  £315, 
when  £1  at  60  days  is  worth  $4.80,  gold  being  quoted 
at  105  ? 

5.  How  large  a  bill  on  London  can  be  bought  for  $2,000 

in  currency,  when  sterling  exchange  is  quoted  at  $4.85 

and  gold  at  106  ? 

Explanation. — Here  we  convert  $4.85  in  gold  into  currency, which 
gives  $4.85  x  1.06  =  $5.14.  Then,  because  it  takes  $5.14  in  currency 
to  buy  £1  of  exchange,  we  have  $2,000  -4-  5.14  =  £389  2s.  Id.    Ana. 

6.  How  large  a  60  day  bill  on  London  can  be  bought  for 
$3,000,  when  60  day  bills  are  quoted  at  $4.84,  and  gold 
at  110  ? 

7.  What  is  the  cost  of  a  bill  on  Paris  for  3,000  francs, 
when  exchange  is  at  the  rate  of  $1  to  ^.16  francs,  and 
gold  is  worth  110  ? 

Ans.  ($5,000  ^  5.15)  x  1.10  =  $1,067,96^. 

8.  What  is  the  cost  of  a  draft  on  Paris  for  30,000  francs, 
when  exchange  is  at  the  rate  of  $1  for  5.25  francs,  and 
gold  is  worth  106  ? 

9.  How  large  a  bill  on  Paris  can  be  bought  for  $1,000  in 
gold,  when  exchange  is  at  the  rate  of  $1  to  5.20  francs  ? 


248  PERCENTAGE     AKD     ITS     APPLICATIONS. 

REVIEW     QUESTIONS. 

(219.)  What  is  exchange?  (220.)  What  is  a  draft  or  bill  of 
exchange?  A  sight  draft?  A  time  draft?  (221.)  What  is  the 
drawer  ?  The  drawee  ?  The  payee  ?  A  remitter  or  buyer  ?  An 
indorser?  A  holder?  (222.)  What  is  an  acceptance?  How  made? 
An  acceptor?  (223.)  What  is  an  inland  bill?  (224.)  Define 
par  of  exchange.  Illustrate.  (225.)  What  is  the  course  of  ex- 
change? Illustrate.  (226.)  What  is  inland  or  domestic  exchange? 
How  are  problems  in  inland  exchange  solved?  (228.)  What  is 
foreign  exchange  ?    (229.)  How  is  foreign  exchange  computed  ? 


XI.     EQUATION    OF    PAYMENTS. 

DEFINITIONS. 

230.  Equation  of  Payments  is  the  operation  of 

finding  the  time  at  which  several  debts  due  at  different 

times  may  be  paid  without  loss  of  interest  to  either  party. 

The  time  at  which  the  payment  may  be  made  is  called  the  Equated 
Time,  and  the  corresponding  date  is  called  the  Equated  Date. 

231.  The  time  that  a  debt  has  to  run  is  called  its 
Time  of  Credit,  and  the  date  from  which  this  time  is 
reckoned  is  called  the  Initial  Date. 

A  credit  of  II  for  a  unit  of  time  is  called  a  Unit  of 
Credit ;  the  product  of  a  debt  by  its  time  of  credit  is  its 
Amount  of  Credit. 

The  unit  of  time  may  be  1  day,  or  1  month,  but  it  must  always  be 
the  same  throughout  the  same  problem. 

232.  The  solution  of  every  problem  in  Equation  of 
Payments  depends  on  the  following 

PRINCIPLE. 

Tlie  amount  of  credit  of  the  sum  of  several  debts  is  equal 
to  the  sum  of  the  amounts  of  credit  of  the  debts  taken 
separately. 


OPERATION. 

400 

X    9    =    3600 

800 

X    6    =    4800 

600 

X    4    =    2400 

EQUATIOI^     OF     PA-XMENTS.  249 

There  may  be  three  cases  :  V,  the  debts  may  be  on  one  side,  and 
have  the  same  initial  date  ;  2'',  the  debts  may  be  on  one  side,  and 
have  different  initial  dates  ;  3°,  some  of  the  debts  may  be  on  one 
side,  and  some  on  the  other,  the  dates  of  payment  being  different. 

233.  When  the  Debts  are  on  one  side  and 
have  the  same  Initial  Date. 

On  the  1st  of  January,  1877,  A.  owes  B.  1400  payable 
in  9  months,  $800  payable  in  G  months,  and  $600  payable 
in  4  months ;  at  what  date  may  the  whole  debt  be  paid 
without  loss  to  either  party  ? 

Explanation.— In 
this  case  the  unit  of 
credit  is  $1  for  1 
month.  Multiplying 
the  different  sums  by 
their  terms  of  credit  1800  )  10800  (  6 

and  adding  the  pro-      A..^  \  Equated  time  =  6  months, 
ducts,   we    find    the  '  ]  Equated  date,  July  1, 1874. 

sum  of  the  amounts 

of  credit  equal  to  10,800  units  ;  but  this  is  equal  to  the  amount  of 
credit  of  $1,800  for  the  required  time  ;  hence,  10,800  divided  by 
1,800  will  give  the  number  of  months  in  the  equated  time.  Adding 
this  to  the  initial  date  we  have  the  equated  date. 

Since  all  similar  cases  may  be  treated  in  the  same  manner,  we 
have  the  following 

RULE. 

/.  Multiply  each  debt  by  its  time  of  credit  and 
find  the  sum  of  the  products;  divide  this  by  the 
sum  of  the  debts  and  the  quotient  will  be  the 
equated  time. 

II.  Add  the  equated  time  to  the  initial  date  and 
the  result  will  be  the  equated  date* 


250  PERCENTAGE     AKD     ITS     APPLICATIONS. 

EXAMPLES. 

1.  A  merchant  owes  $2,400,  of  which  $400  is  payable  in 
6  mos.,  1800  in  10  mos.,  and  $1,200  in  16  7nos.;  what  is  the 
equated  time?  Ans.  12^ months. 

2.  A.  owes  B.  $2,400,  of  which  $800  is  payable  in  6 
months,  $600  in  8  months,  and  $1,000  in  12  months; 
what  is  the  equated  time  ?  Ans.  9  months. 

3.  A  merchant  owes  $300,  payable  as  follows :  $80  in 
22  days,  $100  in  60  days,  and  $120  in  75  days;  what  is  the 
equated  time  ?  Ans.  56  days. 

Note. — The  rule  gives  55f|  days.  In  such  cases  we  shall  neglect 
the  fraction  when  it  is  less  than  ^,  and  add  1  to  the  whole  number 
when  it  is  equal  to,  or  greater  than  ^.  If  the  unit  of  time  is  1  month, 
the  fractional  part  will  be  reduced  to  days  at  the  rate  of  30  days  per 
month,  and  fractional  parts  of  the  latter  will  be  treated  as  just 
explained. 

4.  A.  owes  B.  $1,000,  payable  as  follows:  $200  at 
present,  $400  at  6  months,  and  $400  at  15  months ;  what 
is  the  equated  time  ?  Ans.  8  months  12  days. 

5.  A.  owes  a  sum  of  money,  of  which  \  is  payable  at 
30  days,  ^  at  60  days,  and  the  rest  at  90  days ;  what  is  the 
equated  time  for  the  payment  of  the  whole  ? 

234.  When  the  Debts  are  on  one  side,  and 
have  different  Initial  Dates. 

A.  sells  goods  to  B.  on  a  credit  of  60  days,  as  follows : 
Jan.      14,  1874,  a  bill  of  $2,000 
Feb.      10,     ''     ''   ''   "  $1,500; 
March  25,     "     ''    "    ''  $3,1)00 

Kequired  the  equated  time  and  date  of  payment. 


EQUATION     OP     PAYMENTS.  ^1 

Explanation. — The  operation. 

first    payment    is    due      March  15, 2000  X    0=   00000 
March  15,   the   second     April    11,1500x27=   40500         "^ 
April  11,  and  the  third      ^  24,3000x70=210000 

May  24.    Assuming  the  "^ 

date  of  the  earliest  pay-  6500)  250500(38.538 

ment  as  the  initial  date,  j  Equated  time,  39  days. 

theotherpaymexitswill  ^^-  "j  Equated  date,  April  23,  1874. 
have  27  and  70  days  to  >      r  ^ 

run.  Proceeding  as  before,  we  find  the  equated  time  greater  than 
38.5  days  ;  calling  it  39  days,  according  to  the  rule,  and  adding  it  to 
the  assumed  initial  date,  we  find  the  equated  date  to  be  March 
15  +  39(^a.rr  April  23. 

Since  all  similar  cases  may  be  treated  in  the  same  manner,  we 
have  the  foUowmg 

RULE. 

Find  the  date  of  each  payment,  and  assume  the 

earliest  as  an  initial  date ;  then  find  the  time  of 

credit  of  each  item,  and  proceed  as  in  the  last  case. 

EXAMPLES. 

1.  A.  bought  goods  as  follows  :  May  5, 1500  on  4  months' 

credit ;   May  25,  $750  on  4  months'  credit ;  June  27,  $800 

on  6  months'  credit ;  what  is  the  equated  date  of  payment 

of  the  whole  debt?  Ans.  Oct.  26. 

Note. — When  credit  is  given  by  months,  calendar  months  are 
understood,  without  reference  to  the  number  of  days  they  contain  ; 
in  applying  the  preceding  rule,  the  times  of  credit,  if  not  exact 
months,  are  reduced  to  days,  counting  the  actual  number  of  days  in 
each  month.  Thus,  in  the  example  just  given,  the  items  mature 
Sept.  5,  Sept.  25,  and  Dec.  27,  and  the  times  of  credit,  counting  from 
Sept.  5  as  the  initial  date,  are  0,  20,  and  113  days. 

2.  There  are  three  notes  payable  as  follows:  the  Ji7'st 
for  1500,  Feb.  12,  1877  ;  the  second  for  $400,  March  12, 
1877;  and  the  third  for  $300,  April  1,  1877;  what  is  the 
equated  date  for  the  payment  of  all  ?    A ns.  March  5,1877. 


252  PERCEKTAGE     AND     ITS    APPLICATIONS. 

235.  When  each  party  owes  the  other,  the 
times  of  payment  being  different. 

The  difference  of  the  sums  of  the  deUts  and  credits  of 
an  account  is  called  the  Balance  of  Account. 

The  balance  must  be  added  to  the  smaller  sum. 

By  preceding  rules,  all  the  items  on  either  side  may  be  reduced 
to  a  single  item,  payable  at  a  specified  time.  Hence,  every  account 
may  be  reduced  to  two  items,  one  on  eacli  side. 

The  following  suppositions  illustrate  every  case  that  can 
arise : 

1°.  Suppose  that  A.  owes  B.  1500  payable  July  16, 1877, 
and  that  B.  owes  A.  1200  payable  July  1,  1877. 

2°.  Suppose  that  A.  owes  B.  1200  payable  July  16, 1877, 
and  that  B.  owes  A.  $500  payable  July  1,  1877. 

In  both  cases  it  is  required  to  find  the  date  at  which  the  'balance 
of  account  may  be  paid  without  loss  to  either  party. 

In  each  example  let  the  date  of  the  earlier  payment, 

July  1, 1877,  be  taken  as  the  initial  date. 

1°.  Explanation. — On  the  1st  operation. 

of  July,  A.  was  entitled  to  500  x  15,  5qq  >^  j^g  __  750O 

or  7,500,  units  of  credit ;  but  at  that  7500 

time  the  balance  due  B.  was  but  =  25. 

$300  ;  hence,  the  time  before  the  ^^^ 

balance  became  due  was  3^^^,  or  Ans.  July  26,  1877. 

25  days  ;  adding  this  to  the  initial 

date,  we  find  that  the  balance  was  due  July  36,  1877. 

2°.  Explanation.— On  the  1st 
of  July,  A.  was  entitled  to  200  x  15, 

or  3,000,  units  of  credit ;  but  B.  was  ^^0  X  15  =  3000 

also  entitled  to  the  same  amount ;  3000 

hence,  the  balance  due  him,  $300,  oqq   =  1^^* 

was  entitled  to  credit  for  ^V(r»  or  a        j         01    10*7  »v 

10  days  before  the  1st  of  July,  that  ^^^-  ^^^"6  4,1,  18  77. 

is,  from  the  21st  of  June,  1877. 


EQUATION     OF     PAYMENTS.  353 

Since  all  similar  cases  can  be  treated  in  the  same  manner,  we 
have  the  following 

RULE. 

/.  Reduce  each  side  to  a  single  term,  and  take 
the  date  of  the  earlier  payment  as  an  initial 
date ;  then  multiply  the  side  of  the  account  that 
falls  due  last  by  the  time  between  the  dates  of 
payment  and  divide  the  product  by  the  balance 
of  the  account;  the  quotient  will  be  the  equated 
time. 

II.  If  the  greater  side  of  the  account  falls  due 
last,  add  the  equated  time  to  the  initial  date; 
if  the  smaller  side  falls  due  last,  subtract  the 
equated  time  from  the  initial  date;  the  result 
will  be  the  equated  date  at  which  the  balance  is 
payable. 

EXAM  PLES. 

1.  A.  owes  B.  $8,750  payable  July  21,  1877,  and  B.  owes 
A.  $6,500  payable  June  9,  1877;  when,  and  to  whom,  is 
the  balance  payable  ?  Ans.  To  B.  Nov.  19,  1877. 

2.  A.  owes  B.  $6,500  payable  July  21,  1877,  and  B.  owes 
A.  18,750,  payable  June  9,  1877;  to  whom  is  the  balance 
payable,  and  what  is  its  equated  date  ? 

Ans.  To  A.,  and  the  equated  date  is  Feb.  8, 1877. 

EQUATION    OF    ACCOUNTS. 

236.  Equation  of  Accounts  is  the  operation  of 
finding  the  time  at  which  the  balance  of  an  account  should 
be  made  payable  in  order  that  there  may  be  no  loss  of  in- 
terest to  either  party.  This  time  is  called  the  Equated 
Time. 


:e54 


PERCENTAGE     AND     ITS    4PPLICATIONS. 


237.  Let  it  be  required  to  find  the  equated  time  of 
payment  of  the  balance  of  the  following  account : 

Dr.    Hekry  Ahl  in  acct.  with  Benj.  Barrol.    Or. 


1878. 

1873. 



— 

April  1. 

Merchandise. 

$875  00 

April  20. 

Cash.     .     . 

$500 

00 

"    18. 

" 

250  00 

May    20. 

<j 

185 

00 

May  25. 

'* 

150  00 

ExPLANATiON.^The  first  side  of  the  account  is  equivalent  to  a 
single  item  of  $775  payable  April  17,  and  the  second  side  is  equiv- 
alent to  $685  payable  April  28  (Art.  233). 

By  the  rule  of  Art.  235,  we  find  that  the  balance  is  payable  84 
days  preceding  April  17,  that  is,  on  the  28d  of  January  ;  hence,  the 
equated  date  of  the  balance  is  January  28,  1873. 

All  similar  cases  may  be  treated  in  like  manner ;  hence  the  fol- 
lowing 

RULE. 

Equate  each  side  separately  by  the  rule  of  Art. 

234 ;   then  find  the  equated  time  of  the  balance 

by  the  rule  of  Art.  235. 

EXAMPLE. 

1.  Find  the  equated  date  of  payment  of  the  balance  in 
the  following  account : 


Dr. 


A.  in  acct.  with  B. 


Or. 


1874. 
May     1. 
June  15. 
Oct.    15. 

To  corn  .     . 
"   wheat     . 
"   corn  .     . 

— i-S-i— 1 

1874. 
June  1. 
July   1. 
Nov.  1. 

By  cash  .     . 
If      <« 

$300  00 

1500  00 

1000,00 

$2800  00 

Ans.  April  35,  1874. 


EQUATION    OF    PAYMENTS, 


255 


CASH,  AND  INTEREST  BALANCE. 

338.  The  Cash  Balance  of  an  account  is  the  differ- 
ence between  the  two  sides  of  an  account  with  interest  on 
each  item  to  the  date  of  settlement. 

The  Interest  Balance  is  the  difference  between  the 
total  interest  on  the  items  of  the  two  sides. 

From  these  definitions  we  have  the  following 

RULE. 
Add  to  each  side  of  the  account  the  total  inter- 
est on  all  the  items   of  that  side  to  the  date  of 
settlement ;  the  difference  between  the  footings  of 
the  sides  so  increased  is  the  cash  balance. 

EXA  M  PLES. 

1.  Find  the  cash  balance  of  the  following  account  on 
the  15th  of  August,  1874,  interest  being  at  6^: 

Dr.    Jno.  Irving,  in  met.  luith  Henry  Hour.     Or. 


1874.     1 

1874. 

April    1.    To  mdse. 

$25  00 

May  10. 

By  cash    . 

$30 

00 

'•    20.      "       "     .     . 

1800 

Aug.  12. 

«          a 

35 

00 

June  15.      "       "     .     . 

40 

00 

*'    15. 

Cash  bal. 

18 

822 

Aug.  15.    Bal.  of  Int.  . 

822 

$83 

$83 

822| 

822 

Explanation.  —The  debit  side  of  the  account  is  entitled  to  inter- 
est on  $25  for  13G  days,  on  $18  for  117  days,  and  on  $40  for  61  days, 
all  of  which  is  equivalent  to  the  interest  on  $1  for  7,946  days  ;  in 
like  manner,  the  credit  side  is  entitled  to  interest  on  $1  for  3,015 
days.  The  debit  side  is  therefore  entitled  to  a  balance  of  interest 
equal  to  that  on  $1  for  4,931  days,  or  to  $.822.  We  add  this  to  the 
debit  side  and  then  find  that  the  sum  of  the  items  on  that  side  ex- 
ceeds the  sum  of  the  items  on  the  credit  side  by  $18,822.  This  is 
the  cash  balance  and  is  to  be  added  to  the  credit  side. 


256 


PEKCENTAGE    AND    ITS    APPLICATIONS. 


The  interest  on  $1  for  4,931  days  is  found  by  the  rule  for  days ; 
that  is,  we  divide  4,931  by  6,000,  and  the  quotient  is  the  number  of 
dollars  in  the  required  interest. 

2.  Find  the  cash  balance  of  the  following  account  on 
March  20, 1873,  interest  being  computed  at  7%: 


Br, 


0.  in  acct.  with  D. 


Cr. 


1873. 

1873. 

Feb.     5. 

Tomdse.  .     . 

$680 

00 

Jan.  25. 

To  cash  .     . 

$350'00 

"      34. 

t(               (C 

300 

00 

Feb.  15. 

((      « 

600  00 

Mar.     1. 

IC                 tt 

150 

00 

• 

''     16. 

€(             it 

600 

00 

Ans,  1881.631. 

REVIEW    QUESTIONS. 

(230.)  What  is  equation  of  payments?  The  equated  time? 
The  equated  date?  (231.)  What  is  the  time  of  credit?  The 
initial  date?  What  is  a  unit  of  credit?  The  amount  of  credit? 
(232.)  On  what  principle  does  equation  of  payments  depend? 
How  many  cases  may  arise,  and  what  are  they  ?  (233.)  What  is 
the  rule  when  the  debts  are  on  one  side  and  have  the  same  initial 
date  ?  (234.)  Rule  when  the  debts  are  on  both  sides  and  have  dif- 
ferent initial  dates  ?  (235.)  Rule  when  each  party  owes  the  other, 
the  times  of  payment  being  different?  (236.)  What  is  equation 
of  accounts?  Rule?  (238.)  What  is  the  cash  balance?  The 
interest  balance  ?    Rule  for  finding  cash  balance  ? 


XII.    CUSTOM-HOUSE    BUSINESS. 

DEFINITIONS. 

239.  Duties  are  taxes  laid  by  the  United  States  gov- 
ernment on  certain  kinds  of  imported  goods. 

The  ports  at  which  foreign  goods  may  be  landed  are  called  Ports 
of  Entry.  At  each  port  of  entry  is  a  mstom-hotiae,  at  which  the 
taxes  are  collected. 


CUSTOM-HOUSE     BUSINESS.  257 

240.  Duties  are  of  two  kinds,  specific  and  ad  valorem. 
A  Specific  Duty  is  a  definite  tax  laid  on  a  certain  article 
without  reference  to  its  cost.  An  Ad  Valorem  Duty  is 
a  percentage  on  the  invoiced  price  of  an  article  at  the 
place  from  which  it  is  imported. 

241.  An  Invoice  or  Manifest  is  an  inventory  of 
goods  setting  forth  their  cost  price  at  the  place  from  which 
they  are  imported. 

242.  A  Tonnage  Duty  is  a  tax  laid  on  a  vessel  for 

the  privilege  of  entering  a  port.     This  tax  depends  on  the 

size,  or  tonnage,  of  the  vessel,  the  ton  being  equal  to  a 

capacity  of  40  cubic  feet. 

Weiglits  of  articles  are  generally  expressed  in  tons  of  3,240  lbs. 
The  weight  of  an  article,  before  any  allowance  is  made,  is  called  its 
Gross  Weight ;  its  weight,  after  deducting  all  allowances,  is  called 
its  Net  Weight. 

243.  In  computing  specific  duties  certain  allowances 
are  made  for  losses  incidental  to  commerce. 

Draft  is  an  allowance  for  general  waste ;  it  is  deducted 
before  any  other  allowance  is  made.  Tare  is  an  allow- 
ance for  the  weight  of  the  box,  or  case,  containing  goods. 
Leakage  is  an  allowance  for  loss  on  liquors  imported  in 
casks.  Breakage  is  an  allowance  for  loss  on  liquors 
imported  in  bottles. 

EXAMPLES. 

1.  What  is  the  net  weight  of  176  hJids.  of  sugar,  each 

weighing  10  cwt  1  qr.  14  Ihs.,  draft  being  7  lbs.  per  civt. 

and  tare  56  Its.  per  Tihd.  ?     Ans.  1,623  cwt.  3  qrs.  14  lis. 

Explanation. — In  computing  the  weight,  it  is  to  be  rememhered 
that  28^65.  make  Iqr.,  4:  qrs.  make  Icwt,  and  2Qcwt.  make  \ton. 
In  this  case  the  gross  weight  is  l,^%Qcwt.\  the  draft  is  114ao^.  Wbs.; 
and  the  tare  is  88  cwt, 
9 


258  PERCENTAGE    AN^D     ITS     APPLICATIOKS. 

2.  What  is  the  net  weight  of  100  tierces  of  rice,  each 
weighing  250  lbs.,  tare  being  5%  ?  Ans.  23,750  lbs. 

3.  A  merchant  imported  goods  invoiced  as  follows : 

SOO yds.  cloth  @  13s.  per  yd.; 

873  yds.  carpeting  @  4s.  per  yd. ; 

150  doz.  cambric  hdkfs.  @  2s.  Sd.  per  dozen. 
The  duty  on  the  cloth  was  30^ ;  on  the  carpeting  25^ ; 
and  on  the  hdkfs.  33^;  what  was  the  amount  of  duty  in 
dollars,  allowing  14.8665  to  the  pound  sterling  ? 

Ans.   1529.232. 

4.  A  wine  merchant  imported  wines  as  follows : 

75  baskets  champagne  @  $12  per  basket ; 

18  casks  Madeira  @  $45  per  cask ; 

12  casks  sherry  @  135  per  cask. 
Allowing  4:%  for  leakage  on  the  wine  in  casks,  what  is  the 
amount  of  duty  @  45^  ?  Ans.    1936.36. 

5.  What  is  the  duty  on  54  T.  13  cwt  3  qrs.  20  lbs.  of 
iron,  invoiced  at  145  per  ton,  the  duty  being  33^^^  ? 

Ans.   $820,446. 

REVIEW     QUESTIONS. 

(239.)  What  are  duties  ?  A  port  of  entry?  (240.)  What  is  a 
specific  duty ?  An  ad  valorem  duty?  (241.)  Wliat  is  an  invoice 
or  manifest?  (242.)  What  is  tonnage  duty?  Gross  weight? 
Net  weight?  (243.)  What  is  draft ?  Tare?  Leakage?  Break 
age? 


^10  Pfm  p^opo^iJoiM: 


I  .      RATIO. 

DEFINITIONS. 

t244.  The  Ratio  of  one  number  to 
another   is   the  quotient  of   the   second 
.;      number  divided  by  the  first.     Thus,  the 
ratio  of  3  to  15  is  15  -r-  3,  or  5. 

The  first  number  is  called  the  Antecedent,  the  second 
number  is  called  the  Consequent,  and  both  are  called 
Terms  of  the  ratio.  Thus,  in  the  ratio  just  given,  3  is 
the  antecedent,  15  is  the  consequent,  and  both  3  and  15 
are  terms  of  the  ratio. 


MENTAL     EXERCISES. 

1.  How  many  quarts  in  a  gallon  ?  What  is  the  quotient 
of  1  gallon  divided  by  1  quart  ?  In  this  case,  what  is  the 
antecedent  ?    The  consequent  ?     The  ratio  ? 

Explanation. — In  measuring  anything,  the  unit  of  measure, 
which  is  supposed  to  be  known,  is  the  antecedent ;  the  thing  to  be 
measured,  or  expressed  in  terms  of  this  unit,  is  the  consequent ;  and 
the  number  of  times  that  the  antecedent  is  contained  in  the  conse- 
quent is  the  ratio. 

2.  If  we  take  3  inches  as  a  unit,  what  is  the  measure  of 
1  foot  ?    How  many  times  3  inches  in  1  foot  ?    What 


260  RATIO     AND     PROPORTION. 

then  is  the  ratio  of  3  inches  to  1  foot  ?  In  this  case,  what 
is  the  antecedent  ?  What  is  the  consequent  ?  What  is 
the  ratio  ? 

3.  How  many  times  is  36*.  4r/.  contained  in  £1  ?  What 
is  the  ration  of  35.  4:d.  to  £1  ?  What  is  the  antecedent, 
or  unit  ?  What  is  the  consequent,  or  measure  ?  What  is 
the  ratio  ? 

METHODS     OF     EXPRESSING     A      RATIO. 

24:5.  A  ratio  may  be  written  in  the  form  of  a  fraction, 

or  it  may  be  expressed  by  writing  the  consequent  after  the 

antecedent  with  a  colon  between  them.    Thus,  the  ratio 

of  2  to  4  may  be  written  f ,  or  it  may  be  written  2:4;  in 

either  case  it  may  be  read :  the  ratio  of  2  to  4,  or,  2  is  to  4. 

Note. — Because  a  ratio  is  a  fraction  we  may  multiply  or  divide 
both  of  its  terms  without  altering  its  value. 

246.  From  the  definition  of  a  ratio,  we  have  the  fol- 
lowing relations : 

1°.  The  ratio  =  The  consequent  -r-  The  antecedent. 
2°.  The  consequent  =  The  antecedent  x  The  ratio. 
3°.  The  antecedent  =  The  consequent  -r-  The  ratio. 

EXAM  PLES 

What  is  the  ratio 

1.  Of  3  to  5  ?  .6.  Of  3^  da.  to  34^  da,? 

2.  Of  7  to  35?  7.  Of  14  to  $120? 

3.  Of  3s.  6d.  to  lUs.  edJ       8.  Of  Uyds.  to  4: yds.? 

4.  Of  14|  lbs.  to  43^  IbsJ      9.  Of  72  cts.  to  9  dsJ 

5.  Of  27i^'?/.  to  6\ibu.?  10.  Of  135.  9d.  to  2s,  9dJ 
What  is  the  consequent  when 

11.  The  antecedent  is  7  and  the  ratio  4  ? 

12.  The  antecedent  is  J  and  the  ratio  ^  ? 

13.  The  antecedent  is  5  lbs.  4  oz.,  and  the  ratio  |  ? 


PROPORTION".  261 

What  is  the  antecedent  when 

14.  The  consequent  is  18  lbs.  6  oz.  and  the  ratio  6  ? 

15.  The  consequent  is  $12.75  and  the  ratio  4.25  ? 

16.  The  consequent  is  160  yds.  and  the  ratio  7J  ? 

REVIEAA^      QUESTIONS, 

(244.)  What  is  the  ratio  of  one  number  to  another  ?  What  is 
the  antecedent  ?  The  consequent  ?  Terms  ?  In  measuring  a  mag- 
nitude, what  is  the  antecedent?  The  consequent?  The  ratio? 
(245.)  Explain  the  different  ways  of  writing  and  reading  a  ratio. 
(246.)  Give  the  value  of  the  ratio,  the  antecedent,  and  the  conse- 
quent each  in  terms  of  the  other  two. 


II.     PROPORTION. 

DEFINITIONS. 

247.  A  Proportion  is  an  expression  of  equality  be- 
tween two  ratios.    Thus,  the  expression 

2  :  5  =  8  :  20, 

is  a  proportion.    It  indicates  that  the  ratio  of  2  to  5  is 
equal  to  the  ratio  of  8  to  20. 

In  writing  a  proportion  the  sign  of  equality  is  usually 
replaced  by  a  double  colon.  Thus,  the  preceding  propor- 
tion may  be  written 

2  :  5  : :  8  :  20. 

It  is  then  read  2  is  to  6  as  S  is  to  20. 

The  first  and  fourth  terms  of  a  proportion  are  called 
Extremes  ;  the  second  and  third  terms  are  called  Means. 
Thus,  in  the  preceding  proportion,  2  and  20  are  the 
extremes  and  5  and  8  are  the  7neans. 


262  RATIO     AND     PROPORTIOK. 

SOLUTION      OF     A     PROPORTION. 

248.  The  Solution  of  a  Proportion  is  the  opera- 
tion of  finding  one  of  its  terms,  when  the  other  three  are 
known.  The  rule  by  which  the  solution  is  performed  is 
called  The  Rule  of  Three. 

PRINCIPLES    USED    IN    SOLVING    A    PROPORTION. 

249.  If  we  have  the  proportion   2:5  : :  8  :  20,   we 

may  write  it  under  the  form  |  =  •^.  If  we  multiply  both 
terms  of  the  first  ratio  by  8  and  both  terms  of  the  second 

ratio  by  2,   (Art.  245),   we  shall  have  r — -  =  — — -. 

^  X  o         /&  X  o 

Now,  these  fractions  are  equal,  and  they  have  equal 
denominators  ;  hence,  their  numerators  are  equal,  that  is, 
5  X  8  =  2  X  20 ;  in  this  case  the  product  of  the  means  is 
equal  to  the  product  of  the  extremes.  But  we  can  reason 
in  like  manner  on  any  proportion ;  hence,  we  have  the 
following  principle : 

1°.  Tlie  product  of  the  means  of  any  proportion  is  equal 
to  the  ptroduct  of  its  extremes. 

From  this  principle  we  have  the  two  following 

2°.  Either  extreme  is  equal  to  the  product  of  the  means 

divided  hy  the  other  extreme. 
3°.  Either  mean  is  equal  to  the  product  of  the  extremes 

divided  hy  the  other  mean. 

250.  In  applying  the  preceding  principles  to  the  solu- 
tion of  proportions,  it  is  found  convenient  to  represent 
the  required  term  by  some  letter,  as  x,  and  the  ratio  into 
which  this  term  enters  is  written  after  the  other. 


PROPORTIOJS^.  263 

EXAMPLES. 

1.  Let  it  be  required  to  solve  the  proportion, 

15  :  45  : :  9  :  x. 

Explanation. — In  this  case  the  required  tenn  is  one  of  the 
extremes  ;  hence,  from  principle  2°,  we  have, 

a;  =  45  X  9  -^  15  =  27.  A718. 

Note. — After  indicating  the  solution,  we  cancel  all  factors  com- 
mon to  both  numerator  and  denominator. 

2.  Let  it  be  required  to  solve  the  proportion, 

17  :  113  : :  56  lbs, :  x.     Ans.  x  =  104:  Ids, 

Note. — If  the  first  two  terms  are  denominate,  we  disregard  their 
common  unit.  Thus,  in  the  last  example,  the  ratio  of  $7  to  $13  is 
the  same  as  the  ratio  of  7  to  3. 

Find  the  value  of  x  in  each  of  the  following  examples : 

3.  U6:$S:',x:4:yds.  13.  25  :  14| : :  £7  10s. : a;. 

4.  2:3::18:.r.  14.  84.50 : 21.12J ::  13a: ic. 

5.  8:32::24:ic.  15.  75  :  4.75  : :  a; :  $10f 

6.  32:18  ::16:ar.  16.  £1^:  £71 ::  7:a;. 

7.  S:4:::i:x.  IZ  12lbs,:d0lbs, ::  $2:x. 

8.  L2:6::a;:L3.  18.  lH  hu, -Ad  bu.  ::  26i:x. 

9.  6  ft. :  Hft.  ::$S:x,  19.  100  ft, :  I  ft.  : :  $150  :  x, 

10.  9  da.:  lb  da.::  £2.1:  X.      20.  36  :  21  ::  $90:a;. 

11.  30ft. :  12ift. : :  $650  :x,    21.  44 :  40  : :  123  :  x. 

12.  28 :  IJ  : :  $140 :  x,  22.  £7 :  £11^  : :  4:yds. :  x. 

RULE     OF     THREE. 

251.  The  Rule  of  Three  is  a  rule  for  finding  from 
tJiree  numbers  a  fourth^  to  which  the  third  shall  have  the 
same  ratio  that  the^rs^  has  to  the  second. 

This  rule  depends  on  the  principles  of  article  249. 


264:  RATIO     AlfD     PROPORTION^. 

OPERATION. 

252.  Let  it  be  required  to  solve  the  following  problem '. 

If  40  yds,  of  cloth  cost  1170,  what  will  64  yds.  cost  ? 

Explanation. — We  first  state  solution. 

tJie  problem;  tliat  is,  we  express  40 yds.  :  64 yds.  li^lHO  :x. 
tlie  conditions  of  the  problem  in  *  i  wq 

tlie  form  of  a  proportion.  /lOMnQfin 

Having  written  x  for  the  fourtJi  4UJlUb»U 

term,  we  write  the  number  having  $272     Ans, 

the  same  unit,  that  is,  $170,  for 

the  third  term.  We  then  consider  whether  the  fourth  term  is 
greater,  or  less  than  the  third ;  here  it  is  plain  that  64  yards  costs 
more  than  40  yards,  that  is,  the  fourth  term  is  greater  than  the 
third  ;  we  therefore  write  the  smaller  of  the  remaining  numbers  for 
the  first  term,  and  the  greater  one  for  the  second  term.  Having 
completed  the  statement,  we  solve  the  resulting  proportion  as 
already  explained. 

Since  all  similar  cases  may  be  treated  In  like  manner,  we  have 
the 

RULE     OF    THREE. 

/.  Denote  the  required  number  by  x,  and  write 

it  for  the  fourth  term  of  a  proportion;  then  write 

the  number  that  has  the  same  unit  for  the  third 

term. 

II.  Consider,  from  the  nature  of  the  questioiv, 
whether  the  fourth  term  will  be  greater,  or  less 
than  the  third,  and  write  the  rejnaining  num- 
bers, in  the  sam^e  relative  order,  for  the  first  and 
second  terms. 

III.  Solve  the  resulting  proportion,  and  the  value 
of  jc  will  be  the  answer. 

EXAMPLES. 

1.  If  25  yards  of  silk  cost  181.25,  what  will  37  yards 
cost  ? 


PKOPORTION.  265 

Explanation. — The  fourth  term  x  stands  for  the  cost  of  ZT  yds. ; 
the  third  term  is  therefore  the  cost  of  25  yds.,  that  is,  $81.35.  Now, 
the  cost  of  37  yds.  is  greater  than  that  of  25  yds. ;  hence,  the  first 
term  must  be  less  then  the  second.  We  have  therefore  the 
following 

Statement.— 25  2/^s. :  Zl  yds. :  :  $81.25  :  x. 

Solving  the  proportion,  we  have  x  =  $120.25.    Ans. 

2.  If  a  man  can  walk  83  miles  in  3  days,  how  far  can 
he  walk  in  11  days  ? 

Statement.— 3  days  :  11  days  : :  84  miles  :  x ; 

•.•  X  =  308  miles.    Ans. 
Note. — The  sign  .*.  stands  for  hence. 

3.  If  it  costs  £2  9s.  6d.  to  travel  198  miles,  how  far  can 
I  travel  for  £8  Os.  lO^d.  ? 

Statement.— £2  9«.  Qd. :  £8  Os.  lO^d. : :  198  m. :  x. 

4.  If  12i lbs.  of  gold  cost  $2,878.75,  what  will  3 oz.  cost? 

5.  If  31  cwt.  1  qr.  14  lbs.  of  sugar  cost  $318.45,  what  will 

1,240  ZZ>5.  cost? 

Note. — If  all  the  numbers  have  the  same  unit,  the  nature  of  the 
question  will  show  which  is  to  be  the  third  term. 

6.  If  a  piece  of  property  worth  $3,250  is  taxed  $35.75, 
what  should  be  the  tax  on  a  house  worth  $17,350  ? 

Explanation. — Here  the  answer  is  to  be  the  tax  on  $17,350 ; 
hence,  the  third  term  must  be  the  tax  on  $3,250. 
Statement.— $3,250  :  $17,350  : :  $35.75  :  x. 

7.  Solve  the  proportion  3  :  4  : :  21 :  ic. 

8.  If  3  pairs  of  socks  cost  $1.41,  what  will  7  pairs  cost  ? 

9.  If  ^  tons  of  hay  will  keep  2  cows  for  the  winter, 
how  many  cows  can  be  kept  on  24f  tons  ? 

10.  If  18|  bags  of  coffee  contain  758  lbs.  8  oz.,  how 
many  bags  are  there  in  12,136  lbs.? 


266  RATIO     AND     PROPORTIOIS". 

11.  How  long  will  it  take  to  travel  1,290  miles,  at  the 
rate  of  306f  miles  in  20|-  days  ? 

12.  If  2^  yds.  3  qrs.  of  carpeting,  1  yard  wide,  will  cover 

a  room,  how  many  yds.  of  carpeting.  If  yds.  wide  will  it 

take  to  cover  the  same  room  ? 

Explanation.— It  will  take  fewer  yards  of  the  latter  width  than 
of  the  former  ;  hence,  the  fourth  term  is  less  than  the  third. 

Statement.— If  ^<?s.  :  1yd.::  2^  yds.  3  qrs. :  x. 

13.  If  12  men  can  build  a  wall  in  20  days,  how  many 

men  would  it  take  to  build  it  in  5  days  ? 

Explanation. — It  requires  more  men  to  do  it  in  5  than  in  20 
days  ;  hence,  the  fourth  term  is  greater  than  the  third. 

Statement.— 5  da. :  20  da.  : ;  12  men  :  a-. 

14.  If  a  piece  of  cloth  20  yards  long  and  f  of  a  yard 
wide  is  required  to  make  a  dress,  what  must  be  the  width 
of  a  piece  12  yards  long  to  make  the  same  dress  ? 

Statement.— 12  :20::^yd.:x. 

15.  In  what  time  can  25  men  do  a  piece  of  work  that 
12  men  can  do  in  3  days  ?. 

Statement. — 25  men  :  12  men  : :  3  days  :  x. 

16.  A.  exchanged  60  yards  of  silk,  worth  S2.40  per  yard, 
for  48  yards  of  velvet;  what  did  the  velvet  cost  per  yard? 

17.  If  30  bushels  of  oats,  at  50  cents  a  bushel,  will  pay 
a  debt,  how  much  barley  must  be  given  to  pay  the  same 
debt,  barley  being  worth  75  cents  a  bushel  ? 

18.  If  42  tons  of  coal  cost  $197.40,  what  will  If  tons 
cost? 

19.  If  a  man  can  do  a  piece  of  work  in  20  days,  work- 
ing 10  hours  per  day,  how  long  will  it  take  him  to  do  the 
same  if  he  works  12  hours  per  day  ? 


PROPORTION-.  267 

20.  If  l^  cords  of  wood  cost  $88f ,  what  will  254}  cu.ft. 
cost? 

21.  If  7^^  barrels  of  apples  cost  $31^-,  what  will  32| 
barrels  cost  ? 

22.  If  2  bit.  Ipk.  of  wheat  cost  $1.93},  how  many  bushels 
can  be  bought  for  $96|  ? 

23.  If  10  bushels  of  coal  cost  25.50 /r.,  what  will  13 
bushels  cost  ? 

24.  If  14  meters  of  cloth  can  be  bought  for  360  fr.,  how 
many  meters  can  be  bought  for  875  fr.  ? 

25.  If  it  cost  $40  to  board  3  men  5  weeks,  what  will  it 
cost  to  board  12  men  10  weeks  ? 

Explanation. — The  board  of  3  men  for  5  weeks  is  the  same  as 
the  board  of  1  man  for  15  weeks,  and  the  board  of  12  men  for  10 
weeks  is  the  same  as  the  board  of  1  man  for  120  weeks  ;  hence,  the 
following 

Statement.— 15  wks.  board  :  120  wks.  board  :  :  $40  :  x. 

26.  If  18  men  men  consume  34  bbls.  of  potatoes  in  135 
days,  how  long  will  it  take  45  men  to  consume  102  bbU.  9 

Explanation. — In  the  first  case  each  man  consumes  f  |  bbl)^.  and 
in  the  second  case  ^£^-  hbls. :  in  the  first  case  the  time  is  135  da.  and 
in  the  second  case  it  is  x  da.  Now,  it  will  take  longer  to, consume 
'£r  ^^'^-  than  to  consume  f|  bbl.,  that  is,  the  fourth  term  is  greater 
than  the  third  ;  hence,  the 

Statement.— f  I  bbl  :  J^^.  jj^i  .  .  135  ^^^  .  ^j 

27.  If  12  boys  pay  12,000  for  1  year's  tuition,  what  must 
14  boys  pay  for  ISJ  months'  tuition  ? 

Statement.— 12  x  12  mo.  :  14  x  18^  mo.  :  :  $2,000  :  x. 

28.  If  it  costs  $7.20  to  transport  18-J  cwt,  5-|-  mi.,  what 
will  it  cost  to  transport  112}  T.,  62|-  miles? 

29.  If  20  men  working  11  hours  a  day  for  30  days  can 


268  KATIO    AND    PROPORTION. 

earn  13,300,  how  much  can  36  men  earn  in  40  days  work- 
ing 10  hours  per  day  ? 

Statement.— 20  .  30  .  11  :  36  .  40  .  10  :  :  $3,300  :  x. 

30.  If  7  men  reap  6  acres  in  12  hours,  how  many  men 

must  be  employed  to  reap  15  acres  in  14  hours  ? 

Statement. — ^^  A.  :  \^A.  :  :  7  men  :  x. 

.'.  a;  =  15  men.     Ans. 

31.  If  14  horses  eat  56  bushels  of  oats  in  16  days,  how 

many  horses  will  it  take  to  eat  120  bushels  in  24  days  ? 

Statement.— 11 6w.  :  ^^^bu.  :  :  lATiorses  :  x. 

.'.  x  =  20  Tiorses.    Ans. 

32.  If  12  horses  will  plow  11  A,  in  5  days,  how  many 
horses  will  be  required  to  plow  33  A.  in  18  days  ? 

33.  If  a  man  can  walk  250  miles  in  9  days  of  12  hours 
each,  how  many  days  of  10  hours  each  would  it  take  him 
to  walk  400  miles.  Ans.  17-^  days. 

DISTRIBUTIVE    PROPORTION    AND    PARTNERSHIP. 

253.  The  rule  of  three  enables  us  to  divide  a  number 
into  parts  proportional  to  two  or  more  given  numbers. 

EXAMPLES. 

1.  Let  $140  be  divided  into  three  parts,  proportional  to 

3,  5,  and  6. 

Explanation. — The  sum  of  the  numbers  to  which  the  parts  are 
propoitional  is  14.  Now.  $140  bears  the  same  relation  to  the  first 
part  that  14  bears  to  3. 


Hence,  14  :  3 
Also,  14  :  5 
And,     14  :.6 

Solving  these  proportions,  we  have, 

1st  part,  $30  ;  3d  part,  $50  ;  3d  part 


:  $140  :  the  first  part. 
:  $140  :  the  second  part. 
:  $140  :  the  third  part. 


PROPOETIOK.  369 

2.  Divide  20  lbs.  4  oz.  into  3  parts  proportional  to  3,  5, 
and  10.     Ans.  3  Ids.  6  oz. ;  5  Tbs.  10  oz.;  and  11  Ihs.  4  oz. 

3.  Divide  $540  among  a  man,  his  wife,  and  three  chil- 
dren, so  that  the  wife  shall  have  twice  as  much  as  each 
child,  and  the  man  twice  as  much  as  his  wife. 

Ans.  Man,  $240 ;  wife,  $120  ;  child,  $60. 

4.  A.  and  B.  start  from  places  150  miles  apart  and  travel 
towards  each  other;  A.  travels  7  mi.  per  hour,,  and  B.  travels 
^mi.  per  hour;  how  far  does  each  travel  before  they  meet ? 

Ans.  A  70  mi. ;  B.  80  mi. 

5.  A.,  B.,  and  0.  enter  into  partnership ;  A.  puts  in  $720, 
B.  puts  in  $340,  and  C.  puts  in  $960 ;  if  they  gain  $505,  how 
much  should  each  receive  ? 

6.  A.  and  B.  buy  goods  to  the  amount  of  $400,  of  which 
A.  pays  $150  and  B.  $250 ;  if  they  lose  $100,  how  much  of 
the  loss  must  each  bear  .^ 

7.  A.,  B.,  and  G.  engage  in  a  speculation  towards  which 
A.  contributes  $480,  B.  $720,  and  0.  $1,200;  if  they  all 
gain  $650,  how  much  does  each  gain  ? 

8.  A  bankrupt  owes  A.  $500,  B,  $750,  C,  $900,  and  D., 
$1,250,  but  his  estate  is  worth  only  $1,020;  what  share 
ought  each  to  receive  ? 

ANA  LYSIS. 

254.  Analysis  is  the  method  of  solving  problems  by 
the  direct  application  of  general  principles,  without  the 
use  of  particular  rules. 

Many  of  the  problems  usually  solved  by  the  rule  of  three  can  be 
solved,  more  expeditiously,  by  analysis.  The  method  of  proceeding 
will  be  shown  best  by  examples. 


270  RATIO    AKD    PROPORTION. 


EXAM  PLES. 

1.  If  34  men  can  build  a  house  in  40  days,  how  long 
will  it  take  12  men  to  build  the  same  house  ? 

Analysis.— It  will  take  operation. 

1  man  34  times  as  long  as         34x40    a.  _  ^^^^  ^^      ^^^^ 
it  will  34  men  ;  lience,  it  12 

will  take  1  man  34  x  40, 

or  1360  days.     But  12  men  can  build  it  in  ^-^  of  tlie  time  that  1  man 
can  ;  lience,  12  men  can  build  it  in  1360  da.-^  12,  or  in  113|  days. 

2.  If  2  ciot  3  qrs.  10  lbs.  of  sugar  cost  $34.20,  how  much 
can  be  bought  for  $75.60  ? 

Analysis.— If  we  operation. 

divide  *34.20  by  285,  «34|0  ^  ^^  ^^^^  ^^  ^  ^^ 

the  quotient,  12  eta.,  285 

will  be  the  cost  of  1  lb.  * wk  gQ 

If  we  divide  $75.60  - ..::,    r    =  630,  numher  of  lbs. 

12  els.  -^ 

=  630  lbs.    Ans. 


by    12  cts.,  the   quo- 
tient will  be  the  re-  .  75.60  X  285  Ibs 
quired     number     of          '  *  34.20 
pounds.     The  entire 
operation  is  indicated  in  the  last  line. 

3.  If  6  men  can  reap  80  acres  in  12  days,  how  many- 
days  will  it  take  25  men  to  reap  200  acres  ? 

Analysis. — If  it  takes  6  men  12  opbbation. 

days  to  reap  80  acres,  it  will  take  n^ 

1  man  6  x  12  days  to  do  the  same  ;  200  -r- 

hence,  1  man  can  reap  80  -r-  (6  x  12)  6  X  12 

acres  in  1  day,  and,  consequently,  ann  v,  f?  sx  1  o 

25  meq  can  reap  25  times  as  much  z= =  74-; 

in  1   day,   that  is,   they  can   reap  "^  ^  ^^ 

80  X  25  ^  (6  X  12)  acres  ;    now,  if  .^   ^  ^  ^^^ 

we  divide  200  acres  by  the  number  *^ 

of    acres    that    25   men    can    reap 

in  one  day,  the  quotient,   7^,  will  be  the  required  number  of 

days. 


OPBKATION. 

1            1 

4+6  = 

5 

12' 

-^ 

■H; 

.  2|days. 

Ans. 

PROPORTION".  271 

4.  A.  can  do  a  piece  of  work  in  4  days,  and  B.  can  do  it 
in  6  days;  how  long  will  it  take  them  to  do  it,  if  they 
work  together  ? 

Explanation.  —  A.  can  do  |  of  the 
work  in  1  day,  and  B.  can  do  \  of  it  in 
the  same  time  ;  hence,  both  together  can 
do  i  +  ^,  or  1%  of  it  in  1  day ;  but  if  they 
can  do  y%  of  it  in  1  day,  they  can  do  -^^ 
of  it  in  \  of  1  day,  and,  consequently,  they 
can  do  ||  of  it  in  ^^-  days,  that  is,  in  2f 
days. 

5.  Three  men  hire  a  pasture  for  $45 ;  the  first  puts  in 
3  horses  for  5  weeks,  the  second  puts  in  4  horses  for  3 
weeks,  and  the  third  puts  in  7  horses  for  4  weeks ;  what 
should  each  man  pay  ? 

Explanation.— Since  operation. 

the  pasturage  of  3  horses      igt.  3  x  5  =  15  ;  fl  of  $45=112^ 

horse  for  15  weeks,  the      3d.    7x4=28;  ff  of  $45 = S22|^ 
first  received  the  benefit  g^^^    55 

of   15  weeks'   pasturage 

for  1  horse  ;  the  second,  in  like  manner,  received  12  weeks'  pas- 
turage for  1  horse  ;  and  the  third  received  28  weeks'  pasturage  for 
1  horse  ;  they  all  received  55  weeks'  pasturage  for  1  horse.  Hence, 
the  first  should  pay  if,  the  second  if,  and  the  third  |f  of  the 
rent. 

In  like  manner  other  problems  may  be  analyzed  and  solved. 
Let  all  the  examples  in  Article  252  be  solved  by  analysis,  and 
also  the  following-: 

6.  If  2iu.  Iph.  of  wheat  cost  $2.43,  what  will  l^hu. 
cost?  Ans.   $15.39. 

7.  If  14  men  can  board  1  week  for  $45.50,  how  long  can 
3  men  board  for  $97.50  ?  Ans.    10  iveeks. 


272  RATIO     AND     PROPORTIOIf. 

8.  If  a  steamer  sails  728  miles  in  2^  days,  how  far  will 
she  sail  in  12 J  days  ?  Aiis.   3,900  7niles. 

9.  If  20  men  perform  a  piece  of  work  in  12  days,  how 
many  men  will  be  required  to  do  a  piece  of  work  3  times 
as  great  in  ^  of  the  time  ?  Ans.   300. 

10.  If  450  lbs,  of  coffee  cost  $99,  how  much  will  1,450  lbs, 
cost?  A71S.  1319. 

11.  If  27  tons  of  iron  cost  $540,  how  much  will  37^ 
tons  cost  ?  Alls.   1750. 

12.  If  17  bushels  of  wheat  are  worth  125.50,  how  much 
are  29  bushels  worth  ?  Ans.  $43.50. 

13.  If  3  dozen  of  wine  cost  128.50,  how  much  will  5| 
dozen  cost  ?  Ans.  152.25. 

14.  If  117  bushels  of  barley  cost  1105.30,  how  much  will 
413  bushels  cost?  Ans.   1371.70. 

15.  If  36  gallons  of  molasses  cost  $32.40,  how  much  can 
be  bought  for  $105.30  ?  A7is,    111  gals. 

16.  If  I  pay  $3.00  for  riding  40  miles  in  a  stage-coach, 
jaow  far  can  I  ride  for  $10.42|-  ?  Ans,   139  miles. 

17.  If  32  acres  of  land  can  be  bought  for  $1,504,  how 
much  can  be  bought  for  $5,546  ?  Ans.  118  acres. 

18.  Find  the  cost  of  25 J  lbs.  of  tea,  when  17  lbs.  can  be 
bought  for  $15.30.  A7is.  $22.95. 

19.  If  11  Irish  miles  are  equal  to  14  English  miles,  what 
is  the  length,  in  English  miles,  of  a  road  that  measures 

57  Irish  miles  ?  Ans.   72 j\  English  miles. 

20.  If  a  staff  4  feet  high  casts  a  shadow  6  feet  long, 
what  must  be  the  height  of  a  pole  that  will  cast  a  shadow 

58  feet  long  at  the  same  time  ?  Ans.  ^^ft. 


PROPORTIOIS".  273 

21.  A  man  paid  $36  to  several  laborers ;  to  each  man 
he  paid  $4,  and  to  each  boy  $2 ;  the  number  of  men  was 
equal  to  the  number  of  boys ;  how  many  were  there  of 
each  ?  Ans.   6. 

22.  If  14  men  consume  120  worth  of  flour  in  15  days, 
how  many  days  will  it  last  21  men  ?  A7is,  10  days. 

23.  If  a  barrel  of  flour  will  make  180  ten-cent  loaves, 
how  many  eight-cent  loaves  will  it  make  ?       Ans.  225. 

24.  A  horse  and  saddle  together  were  worth  $100,  and 
the  horse  was  worth  9  times  as  much  as  the  saddle ;  what 
was  the  horse  alone  worth  ?  Ans.   $90. 

25.  A  farmer  puts  a  flock  of  sheep  in  3  pastures ;  in 
the  first  he  puts  J  of  his  flock,  in  the  second  J  of  his 
flock,  and  in  the  third  he  puts  32  sheep;  how  many 
sheep  has  he  ?  A7is.   192. 

26.  What  number  is  that  to  which  if  its  sixth  part 
and  its  eighth  part  be  added  the  sum  will  be  186  ? 

Ans.   144. 

27.  A  woman  buys  eggs  at  the  rate  of  3  for  5  ds.,  and 
sells  them  at  the  rate  of  4  for  7  cts.,  clearing  9  cts.  by  the 
bargain  ;  how  many  does  she  buy  ?  Ans.   108. 

28.  A  farmer  gave  5  loads  of  straw  for  »12  tons  of  coal, 
worth  $6^  per  ton;  what  did  he  get  per  load  for  his 
straw?  Ans.   $15|. 

29.  At  what  time  between  1  and  2  o'clock  are  the  hands 
of  a  watch  together  ? 

Explanation. — The  hands  are  together  at  12  o'clock,  and  the 
hour  hand  gains  55  minute  spaces  in  an  hour ;  but  it  must  gain 
60  minute  spaces  before  they  can  be  together  again  ;  hence,  the 
time  required  is  |0  hrs.,  or  1  hr.  Q^jmin.,  that  is,  they  are  together 
at  5x\  minutes  past  1.     Ans. 


274  RATIO     AND     PROPORTION. 

30.  If  sugar  is  worth  7^  cts.  per  lb.,  how  many  pounds 
can  be  bought  for  2J  T,  of  iron,  at  $60  per  ton  ? 

Ans.  2,000. 

Note. — Problems  relating  to  the  distribution  of  loss  or  gain 
among  partners  may  be  solved  like  example  5. 

31.  Two  men  enter  into  partnership ;  the  first  puts  in 
160,  the  second  $80,  and  they  gain  $35 ;  what  is  the  gain 
of  each  ?  Ans.   1st,  $15  ;  2d,  $20. 

32.  A.,  B.,  and  0.  enter  into  speculation ;  A.  puts  in 
$4,000,  B.  puts  in  $5,000,  and  C.  puts  in  $6,000 ;  they  lose 
$2,000 ;  what  part  of  the  loss  must  each  bear  ? 

Ans.  A.,  $533.33^;  B.,  $666. 66|;  and  C,  $800. 

REVIE\Ar    QUESTIONS. 

(247.)  What  is  a  proportion?  How  is  a  proportion  written? 
How  read ?  (248.)  What  is  the  solution  of  a  proportion?  (249.) 
What  principles  are  used  in  solving  proportions?  (250-)  By 
what  do  we  represent  the  unknown  term  of  a  proportion  ?  (251.) 
What  is  the  rule  of  three?  .  (252.)  Give  the  rule  of  three. 
(253.)  How  are  problems  in  partnership  solved  1  (254.)  What 
is  analysis  ? 


PO  W^  ERS. 


DEFINITIONS. 


255,  A  Power  is  the  product  of  two  or 
more  equal  factors.  One  of  these  factors  is 
called  the  Root  of  the  power. 

The  product  of  two  equal  factors  is  called  a  second  power,  or 
square  ;  the  product  of  three  equal  factors  is  a  third  power,  or 
cube ;  the  product  of  four  equal  factors  is  a  fourth  power ;  and  so 
on.  Thus,  3  X  3,  or  9,  is  the  square  of  3 ;  3  x  3  x  3,  or  27,  is  the  cube 
of  3,  and  so  on 

256,  The  Exponent  of  a  power  is  a  number  that 
shows  how  many  times  the  root  is  taken  as  a  factor. 

It  is  written  to  the  right  and  above  the  root.  Thus,  in  the  ex- 
pression 3^,  4  is  the  exponent ;  it  indicates  that  3  is  to  be  taken 
4  times  as  a  factor,  that  is,  3*  =  3  x  3  x  3  x  3  =  81. 


INVOLUTION,    OR    RAISING    TO    POWERS. 

257.  Involution  is  the  operation  of  finding  any  power 
of  a  number. 

The  operation  of  involution  is  also  called  raising  to  powers. 

258.  It  follows  from  the  definitions  already  given  that 
we  may  find  any  power  of  a  number  by  the  following 


276  BOOTS. 

RULE. 

Take  the  numher  as  a  factor  as  many  times  as 

there  are  units  in  the  exponent  of  the  power. 

Note. — To  raise  a  simple  fraction  to  any  power,  raise  each  term 
separately  to  that  power. 

EXAM  PLES. 

Kaise  the  following  numbers  to  the  powers  indicated : 

1.  (4)3.  6.  (.09)2.  11.  (1)3. 

2.  (5)*.  7.  (.15)2.  12.  (1)4. 

3.  (14)3.  8.  (2.5)3.  13.  (2^)3. 

4.  (25)2.  9.  (.33)3.  14.  (3^)3. 

5.  (98)2.  10.  (3.4)2.  15.  (4^)4. 

REVIE^AT    QUESTIONS. 

(255.)  What  is  a  power?  A  second  power?  A  third  power? 
(256.)  What  is  an  exponent  ?  How  written  ?  What  does  it  show  ? 
(257.)  What  is  involution?  What  other  name  has  it ?  (258.) 
Rule  for  raising  a  number  to  a  power. 


II.    ROOTS. 

DEFINITIONS. 

^59.  A  Root  of  a  number  is  one  of  its  equal  factors. 

If  a  number  can  be  resolved,  or  separated,  into  two  equal  factors, 
it  is  said  to  be  a  perfect  square  ;  if  it  can  be  resolved  into  three 
equal  factors,  it  is  said  to  be  a  perfect  cube,  and  so  on. 

260.  The  Square  Root  of  a  Number  is  one  of  its 

two  equal  factors.     Thus,  4  is  the  square  root  of  16. 

If  the  number  is  not  a  perfect  square,  its  square  root  is  only 
approximate. 


SQUARE     ROOT.  277 

All  the  perfect  squares  less  than  100,  with  their  square 
roots,  are  written  in  the  following 


TABLE. 

Perfect  squares,  1    4    9     16    25    36    49 

64    81, 

Square  roots,        12     3      4      5       6       7 

8      9. 

Note. — The  sign  4/  ,  called  the  Radical  Sign,  shows  that  the 
square  root  of  the  number  under  it  is  to  be  taken.  Thus,  ^36 
denotes  that  the  square  root  of  36  is  to  be  taken. 

METHOD  OF  EXTRACTING  A  SQUARE  ROOT. 

261.  The  method  of  finding  the  square  root  of  a  num- 
ber depends  on  the  principles  of  Algebra.  (See  Manual 
of  Algebra,  Art.  107.)  In  accordance  with  these  prin- 
ciples, we  have  the  following 

RULE. 
/.  Separate  the  given  number  into  periods  of 
two  figures  each,  beginning  at  the  right  hand; 
the  period  on  the  left  will  often  contain  but  one 
figure. 

II.  Find  the  greatest  perfect  square  in  the  first 
period  on  the  left  and  place  its  square  root  on  the 
right,  after  the  manner  of  a  quotient  in  division  ; 
subtract  the  square  of  this  root  from  the  first 
period,  and>  to  the  remainder  bring  down  the 
second  period  for  a  dividend. 

III.  Double  the  root  found  and  place  it  on  the 
left  for  a  d^ivisor.  See  how  many  times  this 
divisor  is  contained  in  the  dividend,  exclusive 
of  the  right  hand  figure,  and  place  the  quotient 
in  the  root  and  also  at  the  right  of  the  divisor. 


278  ROOTS. 

IV.  Multiply  the  divisor,  thus  augmented,  by  the 
last  figure  of  the  root  already  found,  subtract  the 
product  from  the  dividend  and  to  the  remainder 
bring  down  the  next  period  for  a  new  dividend. 

V.  Double  the  root  already  found  for  a  new 
divisor,  and  continue  as  before,  until  all  the 
periods  have  been  brought  down  and  operated  on. 

Notes. — 1.  If  any  quotient  figure  proves  too  large,  let  it  be 
diminished  until  it  gives  a  product  less  than  the  partial  dividend. 

2.  If  the  last  remainder  is  0,  the  given  a  number  is  a  perfect 
square  and  the  root  is  exact ;  if  not,  the  root  is  true  to  within  less 
than  1. 

3.  The  square  root  of  a  simple  fraction  is  equal  to  the  square 
root  of  its  numerator  divided  by  the  square  root  of  its  denominator, 

EXAMPLES. 

1.  Find  the  square  root  of  8836. 

Explanation. — The  two  periods  are  88  operation. 

and  36  ;   the  greatest  perfect  square  in  88  ftSSfiCQi 

is  81,  (table.  Art.  260),  and  its  square  root  — 

is  9  ;  this  we  write  as  the  first  figure  of  the  "^ 


root  and  place  its  square  81  under  the  first  18/4)73/6 

period  ;  subtracting,  we  have  7  for  a  remain-  no  q 

der,  to  which  we  bring  down  the  period  36  

for  a  dividend  ;    doubling  9  we   have  18,  0 

which  we  place  on  the  left  for  a  divisor,  and 

this  is  contained  4  times  in  73  ;  we  therefore  place  4  on  the  right  of 
9  and  also  on  the  right  of  18 ;  multiplying  184  by  4  we  find  736, 
which  taken  from  736  gives  0  for  a  remainder  ;  hence,  the  squaie 
root  of  8836  is  94. 

Perform  the  following  indicated  operations : 


2.  V9604. 

6.  V14641. 

10. 

A/«i. 

3.  \/T3225. 

7.  V37636. 

11. 

V^TT- 

4.  V342225. 

8.  V41616. 

12. 

VfU. 

5.  V944784. 

9.  V52441. 

13. 

V^T- 

SQUARE     ROOT.  279 

Note. — If  there  is  a  remainder  the  operation  may  be  continued 
by  annexing  periods  of  decimal  ciphers ;  for  each  period  thus 
annexed    there  will   be  one  decimal  figure  in  the  root.      Thus, 

/v/i87  =  ^^187.0000  =  13.67.    Here  the  approximate  root  is  true  to 
within  less  than  .01. 

Find  the  square  roots  of  the  following  numbers  to  two 
decimal  places : 

14.  229.    Ans.  15.13.  16.  450.    Ans,  21.21. 

15.  354.    Ans.  18.81.  17.  592.    Ans.  24.33. 

Note. — The  square  root  of  a  decimal  may  be  found  by  the  pre_ 
ceding  rule.  In  this  case  we  begin  to  point  off  periods  at  the  deci. 
mal  point  and  proceed  toward  the  right.  Any  simple  fraction  may 
be  changed  to  a  decimal  and  then  operated  upon  by  the  rule. 

Find  the  square  roots  of  the  following  numbers  to  three 
places  of  decimals : 

18.  .0249.     Ans.  .157.  21.  .152881.     Ans.  .391. 

19.  .69.    Ans.  .830.  22.  .326041.     Ans.  .571. 

20.  .1051.    Ans.  .324.  23.  .010404.    Ans.  .102. 

PRACTICAL      PROBLEMS. 

1.  A  general  forms  an  army  of  117,649  men  in  a  square; 
how  many  men  are  there  in  each  rank  and  how  many 
ranks  in  the  square  ?  Ans.  343. 

2.  In  a  square  pavement  there  are  48,841  stones,  each 
1ft.  square ;  what  is  the  length  of  the  pavement  and  what 
is  its  breadth?  Ans.  221fL 

3.  A  square  farm  contains  640  acres ;  how  long  is  each 
gide?  Ans.  S20  rods. 

4.  A  square  field  contains  160  acres;  what  will  it  cost 
to  build  a  wall  around  if  each  rod  of  wall  cost  $2  ? 

Ans.  $1,280. 


280  ROOTS. 

CUBE      ROOT. 

262.  The  Cube  Root  of  a  Number  is  one  of  its 

three  equal  factors.     Thus,  5  is  the  cube  root  of  125. 

If  a  number  is  not  a  perfect  cube  its  cube  root  is  only  approx- 
imate. 

All  the  perfect  cubes  less  than  1,000,  with  their  cube 
roots,  are  written  in  the  following 

TABLE. 

Perfect  cubes,    1     8    27     64    125    216     343    512     729; 
Cube  roots,        12345        6        7        8        9. 

Note. — The  sign,  ^  ,  shows  that  the  cube  root  of  the  number 
under  it  is  to  be  taken.  Thus,  /y/125  denotes  that  the  cube  root  of 
125  is  to  be  taken.  The  number  3  written  over  the  sign  is  called  an 
Index. 

METHOD  OF  EXTRACTING  A  CUBE  ROOT. 

263,  The  method  of  finding  the  cube  root  of  a  num- 
ber depends  on  the  principles  of  algebra,  (see  Manual  of 
Algebra,  Art.  111).  In  accordance  with  these  principles 
we  have  the  following 

RULE. 

I.  Separate  the  numher  into  periods  of  three  fig- 
ures each,  heginning  at  the  right;  the  left-hand 
pe7%od  will  often  contain  less  than  three  figures. 

II.  Find  the  greatest  perfect  cube  in  the  first 
period  on  the  left,  and  set  its  root  on  the  right  after 
the  manner  of  a  quotient  in  division;  subtract  the 
cube  of  this  root  from  the  first  peHod  and  to  the 
remainder  bring  down  the  first  figure  of  the  next 
period  for  a  dividend. 


CUBE     ROOT.  281 

III.  Take  three  times  the  square  of  the  root  thus 
found  for  a  divisor,  find  how  many  times  it  is  con- 
tained in  the  dividend,  and  place  the  quotient  for  a 
second  figure  of  the  root.  Cube  the  number  thus 
found,  and,  if  its  cube  is  greater  than  the  first  two 
periods,  diminish  it  successively  by  1  until  its  cube 
is  less  than  the  first  two  periods ;  then  subtract  the 
result  from  the  first  two  periods  and  to  the  remain- 
der bring  down  the  first  figure  of  the  next  period 
for  a  new  dividend. 

IT.  Take  three  times  the  square  of  the  root  found 
for  a  new  divisor  and  proceed  as  before,  continuing 
the  operation  tilt  the  periods  have  been  operated  on. 

Notes. — 1.  If  the  last  remainder  is  0  the  number  is  a  perfect 
cube  and  the  root  is  exact ;  if  not,  the  root  is  true  to  within  less 
than  1. 

3.  The  cube  root  of  a  simple  fraction  is  equal  to  the  cube  root  of 
its  numerator  divided  by  the  cube  root  of  its  denominator. 

3.  The  cube  root  of  a  decimal  or  the  approximate  cube  root  of  an 
Imperfect  cube,  may  be  found  by  a  process  entirely  similar  to  that 
employed  in  finding  the  square  root  in  similar  cases. 

EXAM  PLES. 

1.  Find  the  cube  root  of  804357. 

Explanation.— The  num-  opebation. 

ber  having  been    separated  804  357  (  93 

into    periods,    we    find    the  ^g ^J^ 

greatest  cube  in   804  to  be  ^    —  ^"^^ 

739  and  its  cube  root,  9,  is  3  ^  92  _  243  )  753 

the  first  figure  of  the  root ;  —"oTUq^ 

taking    739    from   804    and  ^^    —  ^^^^^  ^ 

bringing  down  3,  we  have  Q 

753;  dividing  this  by  3  times 

the  square  of  9,  or  243,  we  get  3  for  the  second  figure  of  the  root ; 

cubing  93  we  find  the  result  equal  to  the  given  number  ;  hence  93 

is  the  required  root. 


282  PEOGRESSIONS. 

Perform  the  following  indicated  operations : 

2.  ^531441.    Ans.  81.  6.   ^/W.      Ans,  4.06. 

3.  ^^970299.     Ans.  99.  7.  'V^lot    ^^5.  4.7. 

4.  v^SSM?.      ^ws.  33.  8.   '^206.     Ans.  6.9. 

5.  \^224755712.  AnsMS.        9.   ^1^585.    ^^s.  8.36. 

REVIE^A/■    QUESTIONS. 

(259.)  What  is  a  root  of  a  number?  What  is  a  perfect  square? 
A  perfect  cube  ?  (260.)  What  is  the  square  root  of  a  number  ?  What 
is  the  radical  sign  ?  What  does  it  show  ?  (261 .)  What  is  the  rule 
for  extracting  the  square  root  of  a  number  ?  Of  a  simple  fraction  ? 
How  do  you  find  an  approximate  value  of  a  square  root  ?  How  do 
you  find  the  square  root  of  a  decimal  to  any  number  of  places  ?  Of 
a  simple  fraction  reduced  to  a  decimal  ?  (262.)  What  is  a  cube 
root?  Its  sign?  What  is  an  index?  (263.)  Give  the  rule  for 
extracting  the  cube  root  of  a  number. 


III.    PROGRESSIONS. 

DEFINITIONS. 

264.  A  Progression  is  a  series  of  numbers  that  in- 
crease, or  decrease,  according  to  a  common  law. 

The  numbers  forming  a  progression  are  called  Terms  j  the  first 
and  last  terms  are  called  extremes  and  all  the  rest  are  called  means. 

Note. — Progressions  are  of  two  kinds,  arithmetical  and  geo- 
metrical. 

1°.  ARITHMETICAL    PEOGRESSION. 

DEFINITIONS. 

265.  An  Arithmetical  Progression  is  one  in  which 
each  term,  after  the  first,  is  equal  to  the  preceding  term 
increased,  or  diminished,  by  a  given  number.  This  num- 
ber is  called  the  Common  Difference. 


ARITHMETICAL     PROGRESSION.  283 

If  a  progression  is  formed  by  the  continued  addition  of  a  com- 
mon difference  it  is  increasing ;  if  it  is  formed  by  the  continued 
subtraction  of  a  common  difference  it  is  decreasing. 

The  first  of  the  following  progressions  is  increasing  and  the 
second  is  decreasing : 

2       4       6        8        10 Increasing  progression. 

10        8        6        4         2 Decreasing  progression. 

If  the  increasing  progression  is  inverted,  that  is,  if  it  is  taken  in 
a  reverse  order,  it  becomes  a  decreasing  progression. 

TO    FIND    ANY    T^RM. 

366.  From  the  preceding  definitions  it  is  obvious  that 
we  may  find  any  term  by  the  following 

RULE. 
Multiply  the  commofb  difference  by  the  number 
of  terms  that  precede  the  required  term ;  if  the 
progression  is  increasing,  add  the  product  to  the 
-first  term;  if  the  progression  is  decreasing,  sub- 
tract the  product  from  the  first  term. 

EXAM  PLES. 

1.  The  first  term  of  an  increasing  arithmetical  pro- 
gression is  3,  and  the  common  diiference  is  3  ;  what  is  the 
9th  term  ?  Ans.  3  f  3  x  8  =  27. 

2.  The  first  term  of  a  ^ecre«5m^  arithmetical  progres- 
sion is  36,  and  the  common  difference  is  6 ;  what  is  the 
5th  term  ?  Ans.  36  —  6  x  4  =  12. 

3.  In  an  increasing  progression  the  first  term  is  4,  and 
the  common  difference  is  2 ;  what  is  the  20th  term  ? 

Ans.  42. 

4.  In  a  decreasing  progression  the  first  term  is  45,  and 
the  common  difference  4 ;  what  is  the  8th  term  ? 

Ans.  17. 


OPERATION. 

3, 

6, 

9, 

12,        15,        18 

18, 

15, 

12, 

9,          6,          3 

284  PKOGRESSIOKS. 

TO  FIND  THE  SUM  OF  THE  TERMS. 

267.  A  rule  for  finding,  the  sum  of  the  terms  may  be 
deduced  by  inverting  the  progression  and  proceeding  as  in 
the  following 

OPERATION. 

18  .   .   ,   Given  progression. 
Same  inverted. 

21    +21   +   21    +21+21    +   21  .  .  .  Sum  of  both. 

Explanation. — The  sum  of  the  terms  in  both  progressions  is 
obviously  equal  to  twice  the  sum  of  the  terms  of  the  given  pro- 
gression ;  hence,  the  sum  of  the  terms  of  the  given  progression  is 
-V-  X  6,  or  63. 

Since  all  similar  cases  may  be  treated  in  the  same  manner,  we 
have  the  following 

RULE. 

Multiply  half  the  sum  of  the  extremes  by  the 

number  of  terms. 

EXAM  PLES. 

1.  The  first  term  of  a  progression  is  3,  the  last  term  is 
27,  and  the  number  of  terms  is  9 ;  what  is  the  sum  of  the 
terms  ?     '  Ans.  135. 

Note.— If  the  last  term  is  not  given,  it  may  be  found  by  the 
rule  of  Article  206. 

2.  The  first  term  of  a  decreasing  progression  is  36,  the 
common  difference  is  6,  and  the  number  of  terms  is  5; 
what  is  the  sum  of  the  terms  ? 

Ans,  120. 

3.  In  a  decreasing  progression,  the  first  term  is  45,  the 
common  difference  is  4,  and  the  number  of  terms  is  8; 
what  is  the  sum  of  the  terms  ?  Ans.  248. 


GEOMETRICAL     PROGRESSION^.  285 

4.  What  is  the  sum  of  the  natural  numbers,  1,  2,  3,  &c., 
up  to  99,  inclusive  ?  Ans.  4,950. 

5.  The  first  term  of  a  decreasing  progression  is  15,  the 
last  term  is  5,  and  the  number  of  terms  is  6 ;  what  is  the 
sum  of  the  terms?  Ans.  60. 

6.  The  first  term  of  an  increasing  progression  is  15,  the 
common  dificrence  is  3,  and  the  number  of  terms  is  6; 
what  is  the  sum  of  the  terms  ?  Ans.  135. 

7.  What  is  the  sum  of  the  terms  of  the  progression  1,  2, 
3,  4,  etc.,  up  to  12  inclusive  ?  Ans,  78. 

8.  The  first  term  of  an  increasing  progression  is  7,  the 
common  difference  is  4,  and  the  number  of  terms  is  7; 
what  is  the  sum  of  the  terms  ?  Ans.  133. 

2°.    GEOMETEICAL    PROGRESSION. 

DEFINITIONS. 

268.  A  Geometrical  Progression  is  one  in  which 

each  term,  after  the  first,  is  equal  to  the  preceding  term 

multiplied  by  a  given  number.    This  number  is  called  the 

Ratio  of  the  progression. 

If  the  ratio  is  greater  than  1,  the  progression  is  increadng ;  if 
less  than  1,  the  progression  is  decreasing.    Thus, 

3,  4,  8,  16, 

is  an  increasing  progression,  and 

16,  8,  4,  3. 

is  a  decreasing  progression.  The  ratio  in  the  first  case  is  3  and  in 
the  second  case  it  is  \  ;  in  all  cases,  the  ratio  is  equal  to  the  quotient 
obtained  by  dividing  the  second  term  by  the  first. 

TO    FIND    ANY    TERM. 

269.  In  accordance  with  the  preceding  definitions,  we 
may  find  any  term  by  the  following 


286  PROGRESSIONS. 

RULE. 

Raise  the  ratio  to  a  power  whose  exponent  is  the 
number  of  terms  that  precede  the  required  term 
and  multiply  the  first  term  by  the  result. 

EXAM  PLES. 

1.  In  a  progression  the  first  term  is  3,  and  the  ratio  is  3; 
what  is  the  6th  term  ?  Ans,   3  x  3^  =  729. 

2.  The  first  term  of  a  progression  is  64,  and  the  ratio 
is  i;  what  is  the  5fch  term  ?  Ans.   64  x  (i)^  =  4. 

3.  Find  the  10th  term  of  the  progression  2,  4,  8,  &c. 

A?is.   2  X  29  —  1,024. 

4.  What  is  the  5th  term  of  the  progression  243,  81, 
27,  &c..?  Ans.  3. 

TO  FIND  THE  SUM  OF  THE  TERMS. 

270.  Let  it  be  required  to  find  the  sum  of  4  terms  of 
the  series  2,  8,  &c. 

2  -f  8  +  32  +  128 Indicated  Bum  of  the  terms ; 

8  +  32  4-  128  +  512 4  times  tlie  sum; 

512  —  2 3  times  the  sum; 

512-2       128x4-2       .^_      ^ 

r^: =  i7U. .  .  Required  sum. 

o  o 

Explanation. — Having  indicated  the  sum  of  the  terms,  we  mul- 
tiply each  by  4  and  set  the  products  one  place  toward  the  right  ; 
the  sum  of  these  results  is  4  times  the  sum  of  the  given  series  ; 
subtracting  the  latter  from  the  former,  we  find  512  —  2,  which  is 
3  times  the  required  sum  ;  dividing  by  3,  we  have  170,  which  is  the 
required  sum. 

Since  all  similar  cases  may  be  treated  in  like  manner,  we  have 
the  following 


GEOMETRICAL    PROGRESSIOIS".  287 

RULE, 

Multiply  the  last  term  by  the  ratio;  take  the 
difference  between  the  product  and  the  first  term; 
multiply  this  by  the  difference  between  1  and  the 
ratio. 

EXAMPLES. 

1.  The  first  term  of  a  progression  is  3,  the  last  term 

is   729,  and  the  ratio  is  3  ;    what  is  the  sum  of  the 

terms  ?  .        729  x  3-3       ^  „„„ 

Ans,  T =  1,092. 

Z 

Note. — If  the  first  term  and  ratio  are  given,  the  last  term  may 
be  found  by  the  preceding  rule. 

2.  The  first  term  of  a  progression  is  2,  the  ratio  is  4, 
and  the  number  of  terms  is  5 ;  what  is  the  sum  of  the 
terms?  Ans.  682. 

3.  The  first  term  of  a  geometrical  progression  is  3  and 
the  ratio  is  2;  what  is  the  sum  of  6  terms  ?      Ans.  189. 

4.  The  first  term  of  a  geometrical  progression  is  64  and 
the  ratio  is  J ;  what  is  the  sum  of  6  terms  ?      Ans.  126. 

5.  What  is  the  sum  of  7  terms  of  the  progression  2,  6, 
18,  etc.  ?  Ans.  2,186. 

REVIE^A^     QUESTIONS. 

(264.)  What  is  a  progression?  What  are  terms ?  Extremes? 
Means  ?  (265.)  What  is  an  arithmetical  progression  ?  An  increas- 
ing progression  ?  A  decreasing  progression  ?  (266.)  What  is  the 
rule  for  finding  any  term?  (267.)  For  finding  the  sum  of  the 
terms  ?  (268. )  What  is  a  geometrical  progression  ?  What  is 
the  ratio  ?  When  is  the  progression  increasing  and  when  decreas- 
ing? (269.)  How  do  you  find  any  term?  (270.)  The  sum  of 
any  number  of  terms  ? 


271.  Mensuration  is  the  operation  of 
finding  how  many  times  any  given  magni- 
tude contains  its  unit  of  measure. 

The  unit  of  measure  of  a  magnitude  is  always  a 
magnitude  of  the  same  kind.  The  unit  of  a  line, 
or  the  linear  unit,  is  a  straight  line  of  given  length  ;  as  one  foot :  the 
unit  of  a  surface,  or  the  superficial  unit,  is  a  square  whose  sides  are 
equal  to  the  linear  unit ;  as  one  square  foot :  the  unit  of  a  volume, 
or  the  cubic  unit,  is  a  cube  whose  edges  are  equal  to  the  linear  unit ; 
as  one  cubic  foot. 

Note. — The  rules  for  mensuration  depend  on  the  definitions  and 
principles  of  Geometry,  some  of  which  have  already  been  given. 
In  what  follows,  the  references  are  to  the  Manual  of  Geometry. 
In  these  references  B.  stands  for  Book  and  P.  for  Proposition. 

273.  A  Polygon  is  a  plane  figure  bounded  on  all 

sides  by  straight  lines. 

Each  of  the  bounding  lines  is  called  a  Side  of  the  polygon,  and 
the  point  at  which  any  two  sides  meet  is  called  a  Vertex  of  the 
polygon.  ^^ 

273.  The  Area  of  a  Polygon  is  the 

number  of  superficial  units  that  it  contains. 

274.  A  Triangle  is  a  polygon  of  three 
sides ;  as  ABC ;  the  side  AB  on  which  it 


MENSUBATIOl^. 


289 


EU 


A    K  B 

PARALLELOGRAM. 


E  F 

RECTANGLB. 


is  supposed  to  stand  is  called  its  Base,  and  the  shortest 
distance,  CB,  from  the  opposite  vertex  to  the  base  is  called 
its  Altitude. 

A  Right-angled  Triangle  is  a  triangle  that  has  one 
right  angle.  Thus,  ABC  is  a  right-angled  triangle.  The 
side  AC,  opposite  the  right  angle,  is  called  the  Hypothe- 
nuse. 

*^75.  A  Parallelogram  is  a  polygon  of  four  sides, 
parallel  two  and  two.     A         „ 
Rectangle    is    a    right- 
angled  parallelogram. 

The  figure  ABCD  is  a  par- 
allelogram whose  ham  is  AB 

and  whose  altitude,  or  breadth,  is  KD ;  the  figure  EFGH  is  a  rect- 
angle whose  base  is  EF  and  whose  altitude,  or  breadth,  is  FG. 

276.  A  Trapezoid  is  a  polygon 

of  four  sides,  only  two  of  which  are 

parallel. 

The  figure  ABCD  is  a  trapezoid;  the 
longer  one  of  its  parallel  sides,  is  its  lower 
base,  the  shorter  one  its  upper  base,  and 
the  perpendicular  distance  between  them  is  its  altitude. 

277.  A  Prism  is  a  solid  bounded  by  two  parallel  poly- 
gons called  Bases,  and  by 
parallelograms  called   Lat- 
eral Faces. 

Prisms  are  named  from  their 
bases.  The  figures  in  the  mar- 
gin show  a  quadrangular  prism 
and  a  hexagonal  prism. 


C  E 


F 

TRAPEZOID. 


QUADRANGULAR 
FBISH. 


The  Altitude  of  a  Prism 

is  the  shortest  distance  between  its  bases. 
10 


HEXAQONAIi  PRISM. 


290 


MENSUKATIOJS". 


FBUSTUM. 


278.  A  Pyramid  is  a  solid  bounded  by  a  polygon 
called  the  Base,  and 

by  three  or  more  tri- 
angles called  Lateral 
Faces.  The  lateral 
faces  meet  at  a  point 
which  is  called  the 
Vertex  of  the  pyr- 
amid. 

If  a  pyramid  is  cut  by  a  plane  parallel  to  the  base,  the 
part  included  between  this  plane  and  the  base  is  called  a 
Frustum  of  a  Pyramid.  The  Altitude  of  a  Pyramid 
is  the  shortest  distance  from  the  vertex  to  the  base.  The 
Altitude  of  a  Frustum  is  the  shortest  distance  between 
its  bases. 

The  figure  shows  a  pyramid  and  a  frustum  of  a  pyramid. 

279.  A  Cylinder  is  a  solid  bounded 
by  two  equal  and  parallel  circles  called 
Bases,  and  by  a  curved  surface  called 
the  Convex  Surface. 

The  Altitude   of  a  Cylinder  is  the 

shortest  distance  between  its  bases. 
The  figure  shows  a  cylinder. 

280.  A  Cone  is  a 

solid  bounded  by  a  circle 

called  the  Base,  and  by 

a  curved  surface  called 

the   Convex  Surface. 

The  convex  surface  ta-    £%v., 

pers  uniformly  from  the         ^  conb.  rRusTUM. 


MENSUKATION. 


291 


base  to  a  point  which  is  called  the  Vertex  of  the 
Cone. 

If  a  cone  is  cut  by  a  plane  parallel  to  the  base,  the  part 
included  between  this  plane  and  the  base  is  called  a 
Frustum  of  a  Cone. 

The  Altitude  of  a  Cone  is  the  shortest  distance  from 
the  vertex  to  the  base.  The  Altitude  of  a  Frustum 
is  the  shortest  distance  between  its  bases. 

The  figure  shows  a  cone  and  the  frustum  of  a  cone. 

281.  A  Sphere  is  a  solid  every 
point  of  whose  surface  is  equally  dis- 
tant from  a  point  within  called  The 
Centre. 

A  straight  line  from  the  centre  of  a 
sphere  to  any  point  of  the  surface  is 
called  a  Radius.  A  straight  line  through  the  centre  and 
terminating  at  both  ends  in  the  surface  is  called  a- 
Diameter.  A  plane  through  the  centre  of  a  sphere  cuts 
from  the  sphere  a  Great  Circle.  Any  plane  that  inter- 
sects the  sphere  but  does  not  pass  through  the  centre, 
cuts  from  the  sphere  a  Small  Circle. 

PROPERTY    OF    THE    RIGHT-ANGLED    TRIANGLE. 

282.  It  is  shown  in  geometry 
(B.  4.,  P.  8),  that  the  square  of 
the  hypothenuse  of  a  ri^it-angled 
triangle  is  equal  to  the  sum  of  the 
squares  of  the  other  two  sides. 
Calling  the  sides  about  the  right- 
angle,  the  Base  and  the  Altitude, 
we  have  the  following  relations : 


292  MENSURATION. 


1°.  Hypothenuse  =  ^{Basef  +  {AltUude)\ 
2°.  Base  —  "s/ (Hypoiheniisef  —  {Altitudef. 
3°.  Altitude  =  V {Hypothenusef  —  {Base)\ 


EXAMPLES. 

1.   Find  the   hypothenuse  of  a  right-angled  triangle 
whose  base  is  l^ft.  and  whose  altitude  is  24//. 


Solution.— Hypothenuse  =  ^'(18)^  +  (24)2  ^  39^)5.  ^^5. 

2.  The  hypothenuse  of  a  right-angled  triangle  is  VZ^yds. 
and  its  altitude  is  10  yds. ;  what  is  its  base  ? 


Solution.— Base  =  ^y{l^f  -  (lO)^  =  11^  yds.  Ana. 

3.  The  hypothenuse  of  a  right-angled  triangle  is  ^^ft, 
and  its  base  is  4J//. ;  what  is  its  altitude  ? 


Solution.— Altitude  =  ^/{^f  —  mf  =  6ft.  Ans, 

4.  A  room  is  30//.  long  and  22 j^  ft.  wide ;  what  is  the 
distance  between  two  opposite  corners?         Ans.  37^//. 

5.  A  flag-staff  is  perpendicular  to  a  level  plain  and  a 
rope  71i//.  long  reaches  from  the  top  of  the  staff  to  a 
point  of  the  plain  42f //.  from  the  foot  of  the  staff;  what 
is  the  height  of  the  staff?  Ans.  61! ft. 

6.  A  pair  of  rafters  are  each  22^  ft.  long,  and  the  build- 
ing on  which  they  are  placed  is  36//.  wide ;  how  high  is 
the  ridge  above  the  plane  of  the  eaves  ?         Ans.  IS^ft. 

7.  A.  and  B.  set  out  from  the  same  point  at  the  same 
time ;  A.  travels  due  north  at  the  rate  of  6  miles  an  hour, 
and  B.  travels  due  east  at  the  rate  of  4-J  miles  an  hour; 
how  far  apart  are  they  at  the  end  of  3  hours  ? 

Ans,  22  J  miles. 


MENSURATION.  293 

LENGTH      OF     A     CIRCUMFERENCE. 

283.  It  is  shown  in  Geometry  (B.  5,  P.  11),  that  the 
circumference  of  a  circle  is  equal  its  diameter  multiplied 
by  3.1416;  that  is, 

Circumference  =  Diameter  x  3.1416. 

EXAMPLES. 

1.  What  is  the  length  of  a  circumference  whose  diameter 
is  12  feet  ?  Ans.  12  ft.  x  3.1410  =  37.6992/?f. 

2.  What  is  the  length  of  a  circumference  whose  diameter 
is  6.75 /if.  ?  Ans.  21.8058/A 

3.  Find  the  length  of  a  circumference  whose  radius  is 
8.5  in.  Ans.  53.4072  in. 

4.  What  is  the  circumference  of  a  circle  whose  diameter 
is  20  yards  ?  Ans.  62.832  yds. 

5.  What  is  the  diameter  of  a  circle  whose  circumference 
is  78.54/if.  ?  Ans.  78.54/^.-^3.1416  =  26  ft. 

AREA     OF     A     TRIANGLE. 

284.  It  is  shown  in  Geometry  (B.  4,  P.  4),  that  the 
area  of  a  triangle  is  equal  to  half  the  product  of  its  base 
and  altitude ;  that  is, 

Area  of  triangle  =  Base  x  Altitude -7-2. 

EXAMPLES. 

1.  The  base  of  a  triangle  is  8  feet  and  its  altitude  is 
6  feet ;  what  is  its  area ?  Ans.  24  sq.  ft. 

Note. — By  tlie  term  product  of  tico  lines  we  always  mean  a  rect- 
angle whose  lengtli  is  one  of  the  lines  and  whose  breadth  is  the 
other.    Hence  we  say  that  the  product  of  a  line  by  a  line  is  a  surface. 


294  MENSURATION. 

2.  What  is  the  area  of  a  triangle  whose  base  is  16  yards 
and  whose  altitude  is  3  J  yards  ?  Ans.  28  sq,  yds, 

3.  What  is  the  area  of  a  triangle  whose  base  is  S^  yds. 
and  whose  altitude  is  14  yds.  ?  Ans,  59^  sq.  yds. 

4.  The  area  of  a  triangle  is  74  sq,ft,  and  its  base  is  9^/1. ; 
what  is  its  altitude  ?  -4?i5.  IQft. 

AREA    OF    A    PARALLELOGRAM. 

285.  It  is  shown  in  Geometry  (B.  4,  P.  3),  that  the 
area  of  a  parallelogram  is  equal  to  the  product  of  its  base 
and  altitude;  that  is, 

Area  of  parallelogram  =  Base  x  Altitude. 

EXAM  PLE  s. 

1.  The  base  of  a  parallelogram  is  14  yards  and.  its  alti- 
tude is  5  yards ;  what  is  its  area  ? 

Ans.  14  yds.  x  5  yds.  =  70  sq.  yds. 

2.  Find  the  area  of  a  parallelogram  whose  base  is  13/^. 
and  whose  altitude  is  7J/^.  Ans.  97^  sq.ft. 

3.  A  rectangle  is  7 J-  rds.  long  and  5 J  rds.  wide ;  what  is 
its  area  Ans.  41  J-  sq.  rds. 

4.  A  rectangular  field  contains  4  acres  and  its  length  is 
32  rods  ;  what  is  its  breadth  ?  A71S.  20  rds. 

AREA    OF    A    TRAPEZOID. 

286.  It  is  shown  in  Geometry  (B.  4,  P.  5),  that  the 
area  of  a  trapezoid  is  equal  to  the  half  sum  of  its  bases 
multiplied  by  its  altitude ;  that  is, 

Area  of  trapezoid  =  J  x  (  Upper  dase-\-  Lower  base)  x  Altitude. 


MENSURATIOl^.  295 

EXAMPLES. 

1.  The  parallel  sides  of  a  trapezoid  are  14  yds.  and 
20  yds.  and  the  altitude  is  7  yds. ;  what  is  its  area  ? 

Ans.  ^  of  (14  ^6^5.4-20  yds,)  x  7  yds.  =  119  sq.  yds. 

2.  Find  the  area  of  a  trapezoid  whose  parallel  sides  are 
ISft.  and  22/^.  and  whose  altitude  is  111  ft. 

Ans.  3^0  sq.ft. 

3.  A  board  14/i^.  long  is  18  in.  wide  at  one  end  and  12  in. 
at  the  other  end ;  how  many  square  feet  in  its  area  ? 

Ans.  171  ^Q'  fi' 

4.  The  parallel  sides  of  a  trapezoidal  field  containing 
2J  acres  are  respectively  48  rods  and  32  rods ;  what  is  the 
altitude,  or  breadth,  of  the  field  ?  Ans.  10  rds. 

AREA    OF    A    CIRCLE. 

287.  It  is  shown  in  Geometry  (B.  5,  P.  11),  that  the 
area  of  a  circle  is  equal  to  the  square  of  its  radius  multi- 
plied by  3.1416 ;  that  is. 

Area  of  circle  =  {RadiusY  x  3.1416. 

EXAMPLES. 

1.  What  is  the  area  of  a  circle  whose  radius  is  13  inches  ? 

Ans.  13  in.  x  13  in.  x  3.1416  =530.9304  sq.  in. 

2.  What  is  the  area  of  a  circle  whose  radius  is  2.5  rods  9 

Ans.  19.635  sq.  rds. 

3.  The  radius  of  a  circular  fish-pond  is  75  feet ;  what  is 
its  area  ?  Ans.  17, 671 J  sq.  ft 

4.  The  area  of  a  circle  is  176.715  sq.ft.;  what  is  its 


radius.  Ans.  V176.715-t-3.1416  =  7.5 //J. 


296  MENSURATION. 

SURFACE     OF     A     SPHERE. 

288.  It  is  shown  in  Geometry  (B.  8,  P.  7),  that  the 
surface  of  a  sphere  is  equal  to  4  times  the  square  of  its 
radius  multiplied  by  3.1416;  that  is, 

Surface  of  sphere  =  4  x  {Radius)  ^  x  3. 1416. 

EXAMPLES. 

1.  What  is  the  area  of  the  surface  of  a  sphere  whose 
radius  is  12  inches  ? 

Atis.  4  X 12  in.  x  12  in.  x  3.1416  =  1809.5616  sq.  in. 

2.  Find  the  area  of  the  surface  of  a  sphere  whose  radius 
is  4  feet.  '  Ans.  201.0624  sq.ft. 

3.  A  ten-pin  hall  has  a  surface  of  78.54  sq.  in. ;  what  is 
its  radius  ?     Ans.  '\/78.54  sq.  m.  -^(4  x  3.1410)  =  2 J  in. 

4.  The  radius  of  a  billiard  ball  is  l-J  in.  j  what  is  the 
area  of  its  surface  ?  Ans.  15.904  sq.  in. 

VOLUME,  OR  CONTENT,  OF  A  PA  R  ALL  ELOPI  PE  DO  N, 
PRISM,  OR  CYLINDER. 

289.  It  is  shown  in  Geometry  (B.  7,  Propositions  13 
and  14 ;  B.  8,  P.  1),  that  the  content  of  a  parallelopipedon, 
prism,  or  cylinder,  is  equal  to  the  product  of  its  base  by  its 
altitude ;  that  is,  for  either  of  these  solids  we  have 

Content  =  Base  x  Altitude. 

Note.— The  area  of  the  base  may  often  be  found  by  one  of  the 
preceding  principles. 

EXAMPLES. 

1.  The  base  of  a  parallelopipedon  is  24  sq.  ft.  and  its 
altitude  Sft.;  what  is  its  content?  Ans.  192  cu.ff. 


MEKSURATIOK.  297 

Note.— By  the  term  product  of  a  surface  and  line  we  always 
mean  a  parallelopipedon  whose  base  is  equal  to  the  surface,  and 
whose  altitude  is  equal  to  the  line.  The  number  of  cubic  units  in 
the  volume  is  equal  to  the  number  of  superficial  units  in  the  sur- 
face multiplied  by  the  number  of  linear  units  in  the  height.  Hence, 
we  say,  the  product  of  a  surface  and  line,  or  the  continued  product 
of  three  lines,  is  a  volume. 

2.  The  base  of  a  parallelopipedon  is  81  sq.ft.  and  its 
altitude  4/^, ;  what  is  its  volume  ?  Ans.  324  cu.ft 

3.  Find  the  contents  of  a  prism  whose  base  is  86  sq.ft., 
and  whose  altitude  is  7/if.  Ans.  602  cu.ft. 

4.  The  base  of  a  cylinder  is  equal  to  80  sq.ft.  and  its 
altitude  is  equal  to  ^ft.;  what  is  its  content  ? 

Ans.  4:00  cu.ft. 

5.  The  radius  of  the  base  of  a  cylinder  is  2.5  ft.  and  its 
altitude  is  14/^.;  what  is  its  volume  or  contents  ? 

Ans.  (2.6 ft.)^  X  3.1416  x  14//.=  274.89 cu.ft. 

6.  A  stick  of  hewn  timber  is  27  ft.  long  and  its  cross 
section  is  1.5  sq.ft. ;  what  is  its  content  ? 

Ans.  4:0^  cu.ft, 

CONTENT  OF  A  PYRAMID,  OR  OF  A  CONE. 

390.  It  is  shown  in  Geometry  (B.  7,  P.  17,  and  B.  8, 
P.  2),  that  the  volume  of  a  pyramid  or  of  a  cone  is  equal 
to  the  product  of  its  base  by  ^  of  its  altitude ;  that  is,  for 
either  of  these  solids  we  have 

.     Content  =  Base  X  Altitude  X  i. 

EXAM  PLES. 

1.  A  base  of  a  pyramid  is  49  sq.ft.  and  its  altitude  is  4:  ft. ; 
what  is  its  volume  ? 

Ans.  49  sq.ft.  x  4//.-^3  =  65.3333  cu.ft. 


298  MENSURATION. 

2.  The  base  of  a  cone  is  15.9  sq.  ft.  and  its  altitude  is 
6//. ;  what  is  its  content  ?  Ans.  31.8  cu.ft. 

3.  The  altitude  of  a  cone  is  18/^.  and  the  radius  of  its 
base  is  ^ft;  what  is  its  content?  Ans.  301.5936  cu.ft. 

CONTENT    OF    A    SPHERE. 

391.  It  is  shown  in  Geometry  (B.  8,  P.  8),  that  the 
yolume  or  content  of  a  sphere  is  equal  to  f  times  the 
cube  of  the  radius  multiplied  by  3.1416 ;  that  is, 

Content  of  spliere  =  ^x  {RadiusY  X 3.1416. 

EXAMPLES. 

1.  The  radius  of  a  sphere  is  bft. ;  what  is  its  volume  ? 

.        4x5//^.x5/if.  x5/2f.  X3.1416.      _^  _        ., 
Ans.  ^- — - — ^ =  523.6  cu.ft, 

2.  Find  the  volume  of  a  sphere  whose  radius  is  11.5 /jf. 

Ans.  6370.6412  cw. /if. 

3.  What  is  the  volume  of  a  sphere  whose  radius  is  1\  in.  9 

Ans.  1767.15  cu.  in. 

4.  The  content  of  a  sphere  is  696.9116  cu.  in.  ;  what  is 
its  radius  ?  Ans.  5^  in. 

BOAR  D    M  EASU  RE. 

292.  A  board  foot  is  a  solid  one  foot  long,  one  foot 
wide,  and  one  inch  thick.  It  is  equal  to  one-twelfth  of  a 
cubic  foot. 

This  unit  is  used  in  measuring  boards,  planks,  and  some  kinds 
of  timber. 

Note. — Boards  and  planks  are  of  uniform  thickness  throughout, 
but  they  are  often  of  different  widths  at  the  two  ends  ;  in  this  case 
the  half  sum  of  the  widths  at  the  ends  is  taken  as  the  width  of  the 
board,  or  plank.  The  width  and  thickness  are  usually  expressed 
in  inches,  but  the  length  is  given  in  feet. 


MElTStJRATlON".  299 

The  number  of  board  feet  in  a  board,  plank,  or  stick  of 
timber  may  be  found  by  the  following 

RULE. 
Multiply  the  length  in  feet  by  the  product  of  the 
breadth  and  thickness,  both  in  inches,  and  divide 
the  result  by  12. 

EXAMPLES. 

1.  How  many  board  feet  in  a  board  13  feet  long,  16 
inches  wide,  and  1}  inches  thick  ? 

Ans.  13  X  16  X  li  -T- 12  =  21f. 

2.  A  board  is  1 7  feet  long,  13  inches  wide  at  one  end, 
17  inches  wide  at  the  other  end,  and  1  inch  thick ;  how 
many  board  feet  does  it  contain  ?  A7is.  21  J. 

3.  A  plank  16  feet  long  and  2J  inches  thick  is  16  inches 
wide  at  one  end  and  18  inches  wide  at  the  other;  how 
many  board  feet  does  it  contain  ?  Ans.  56|. 

4.  How  many  board  feet  in  a  piece  of  scantling  18//. 
long,  4  in.  thick,  and  9  i7i.  wide  ?  Ans.  54. 

TIMBER      MEASURE. 

293.  Timber,  when  not  measured  in  board  measure, 
is  usually  measured  in  cubic  feet. 

Timber  may  be  Round,  that  is,  it  may  have  a  circular 
cross  section  ;  or  it  may  be  Hewn,  that  is,  it  may  have  a 
rectangular  cross  section. 

Tlie  cross  section  may  be  the  same  throughout,  or  it 
may  be  greater  at  one  end  than  at  the  other.  The  Mean 
Cross  Section  is  the  cross  section  midway  between  the 
two  ends.  \ 


300  MENSURATION-. 

The  cross  section  of  a  round  stick  of  timber  at  any 
point  can  be  found  when  we  know  its  girt  at  that  point. 
The  Girt  is  the  circumference  after  the  bark  is  removed. 

The  cross  section  in  square  inches  may  be  found  by  the 

RULE. 

Multiply  the  square  of  the  girt  in  inches  by 
.0796. 

EXAMPLES. 

1.  The  girt  of  a  round  stick  of  timber  is  42  inches; 
what  is  its  cross  section  ? 

A71S.  422  X  .0796  =  140.41445^.  in, 

2.  Find  the  cross  section  of  a  round  stick  at  a  point 
where  the  girt  is  52  inches.  Ans.  215.23845^.  in. 

3.  If  the  girt  is  60  inches,  what  is  the  cross  section  ? 

Ans.  286.56  sq.  in. 
The  cross  section   of  a  hewn  stick  of  timber  at  any 
point  may  be  found  by  the  following 

RULE. 

Multiply  the  breadth  of  the  stick  by  its  thickness 
at  that  point,  both  in  inches ;  the  product  is  the  cross 
section  in  square  inches. 

EXAMPLES. 

4.  The  breadth  of  a  square  stick  of  timber  at  its  larger 
end  is  14  inches  and  its  thickness  is  13  inches ;  what  is 
its  greatest  cross  section  ?  Ans.  182  sq.  in. 

5.  The  breadth  of  the  same  stick  at  its  smaller  end  is 
12  inches  and  its  thickness  is  10  inches;  what  is  its 
smallest  cross  section?  Ans.  120 sq.  in. 


MUNSU  RATION".  301 

6.  The  breadth  of  the  same  stick  at  the  middle  of  its 
length  is  13  inches  and  its  thickness  is  11|  inches ;  what 
is  its  mean  cross  section  ?  Ans.  152|  sq.  in. 

Knowing  the  two  end  sections  and  the  mean  section 

in  square  inches,  and  the   length  in  feet,  we  can  find 

the   number  of  cubic  feet  in  a  stick  of  timber  by  the 

following 

RULE. 

To  the  sum  of  the  end  sections  add  four  times 

the  mean  section,  all  in  square  inches,  and  multiply 

the  result  by  the  length  in  feet ;  then  divide  by  86 4> 

and  the  quotient  will  be  the  number  of  cubic  feet 

required. 

EXAM  PLES. 

7.  The  end  sections  of  a  stick  of  timber  are  182  and 
120  55'.  in.,  the  middle  section  is  152^  sq.  in.,  and  its  length 
is  soft. ;  what  is  its  content  ? 

864  -^ 

8.  The  end  sections  of  a  stick  of  timber  4:0ft.  long  are 
46O  and  AOO sq.  in.,  and  the  mean  section  is  440^5'.  in.; 
what  is  its  content  ?  A71S.  121.2963  cw. /if. 

METHOD      OF      DUODECIMALS. 

294.  If  the  linear  dimensions  are  expressed  in  feet  and 
inches,  areas  and  volumes  may  be  found  by  the  Method 
of  Duodecimals. 

In  this  system  of  numbers  the  primary  unit  is  1  foot; 
it  may  be  a  linear  foot,  a  square  foot,  or  a  cubic  foot. 

One  twelfth  of  a  foot  is  called  a  Prime,  one  twelfth  of 
a  prime  is  called  a  Second,  and  one  twelfth  of  a  second 
is  called  a  Third,  as  shown  in  the  following 


302  MENSUKATIOK. 

TABLE. 

12  thirds    '"    make  1  second ". 

12  seconds  "      1  prime , '. 

{ft- 
12  primes  "      1  foot X  sq.ft. 

{  cu.ft. 
The  scale  of  the  system  is  uniform,  that  is,  it  is 
12,  12,  12. 

In  accordance  with  the  principles  laid  down  for  multi- 
plying lines  by  lines,  and  surfaces  by  lines,  w^e  see  that 
feet     multiphed  by  feet       give  feet ; 
feet  "  "    'primes     "    primes; 

feet  "  "    seconds    "     seconds; 

primes       "  "  primes     "     seconds; 

primes       "  "    seconds    "     thirds. 

OPERATION     OF     MULTIPLICATION. 

295,   Let  it  be  required  to  find  the  continued  product 

of  3  ft.  5  in.,  2  ft.  6  in.,  and  4: ft.  7  in. : 

Explanation. — Having  written  the  operation. 

first  two  numbers  so  that   units  of  j^i           /          n         m 

the  same  name    stand   in  the  same  ^  ' 
column,  we  begin  at  the  left  hand 

and   multiply  all    the    parts    of   the  2          6. 

multiplicand  by  2,  writing  the  pro-  g        Tq 
ducts,    without    reduction,    in    their 

proper  columns  according  to  the  prin-         -*-"        ^^ 

ciples   explained   in   the   last  article.  8          6          6 

We  then  multiply  all  the  parts  of  the  a          « 

multiplicand  by  6,  and  place  (the  pro-  

ducts  in  their  proper  columns,  as  de-  32        24        24 

termined  by  the  rules  in  the  last  arti-  56        42        42 

cle.    We  next  add  the  partial  products  '^          ~          ^          ^ 
by  the  rule  for  addition  of  compound 

numbers,  which  gives  8  sq.  ft.  6'  6".  =  39-^  cu.ft. 


MENSURATIOI^.  303 

We  now  multiply  8  sq.  ft.  6'  6"  by  4: ft.  7  in.,  in  the  same  manner 
as  before  and  find  for  the  required  product  39  cu.  ft.  1'  9"  6",  that 
is  (39  +  ^  + Yf^  +  XTaF)>  cu.ft.,  which  is  equal  to  39//^  cu.ft. 

In  like  manner  we  may  multiply  in  all  similar  cases ;  hence,  the 
following 

RULE. 

/.  Wi'ite  the  numbers  so  that  units  of  the  same 
nam^  shall  stand  in  the  same  column. 

II.  Multiply  all  the  parts  of  the  multiplicand 
by  each  part  of  the  multiplier  and  write  the  cor- 
responding partial  products,  ivithout  reduction,  in 
their  proper  columns. 

III.  Add  the  partial  products  by  the  rule  for 
addition  of  cornpound  numbers. 


EXAMPLES. 

Multiplicanc 

ft- 

I....   3 

2 

n 

ft' 
5 

(2.) 

7 

Multiplier . . 

....   5 

7 

7 

10 

15 

10 

35 

49 

21 

14 

50      70 

Product.,.. 

...17 

8 

2 

43 

8      10 

=  ] 

Vmsq. 

ft' 

=  434f.^./if. 

3.  Multiply  Zft  7  in.  by  dft.  4:  in. 

Ans.   33  sq.ft.  5'  4"  =  33^  sq.  ft 

4.  Multiply  6  ft.  ll*?i.  by  16  ft.  2  in. 

Ans.   95  sq.ft.  7'  10"  =  95^  sq.ft. 
6.  Find  the  continued  product  of  3  ft.  4:  in.,  2  ft.  11  in., 
and  6  ft.  11  in.     Ans.  59  cu.ft.  6'  8"  8'"  =  m^\  cu.ft. 


304  MEKSURATION. 

6.  Find  the  continued  product  of  ^ft  3  in,,  hft.  2  in., 
and  6fL  5  in,    Ans.  140  cu,ft  10'  9"  6'"  =  140|f|  ciufL 

PRACTICAL     PROBLEMS. 

1.  How  many  square  feet  in  a  ceiling  17/^.  3  m.  long 
and  11/^.  5  in,  wide  ?  ^7^5.  196||  s^.  /^. 

2.  How  many  square  feet  in  a  pavement   12ft.  6  in. 
long  and  lO/jJ.  2  in.  wide  ?  ^^5.  127^  sq.  ft. 

3.  Find  the  capacity  of  a  box  3/^.  3  in.  long,  2/^.  9  in. 
wide,  and  1/^.  11  in.  deep.  ^W5.  Vl-^cu.ft. 

4.  Find  the  contents  of  a  stick  of  timber  42/^.  6  m 
long,  1  ft.  7  *w.  wide,  and  1  ft.  4  in.  thick. 

5.  What  is  the  capacity  of  a  bin  1ft.  3  m  long,  4/"/.  2  i'^. 
wide,  and  Zft.  5  m.  deep  ?  Ans.  103^  cu.  ft 

6.  How  many  cords  of   wood  in  a  pile   13/^.  3  in, 
long,  4 /if.  2  m.  wide,  and  Zft.  6  in.  high  ? 

Ans.   \C.  65iicu.ft, 

7.  How  many  cords  in  a  pile  20  ft.  4  in.  long,  4//f.  3  in. 
wide,  and  5/^.  2  in.  high  ?  ^;^5.   3  C.  62f|  cz^/if. 

8.  How  many  cubic  feet  of  stone  in  a  wall  27  ft.  6  in, 
long,  3  ft.  3  m.  thick,  and  4:  ft.  2  in.  high  ? 

^/^5.   372i|c«^./^. 

9.  What  is  the  area  of  a  rectangle  whose   length  is 
9  ft.  7  in.,  and  whose  breadth  is  7  ft.  4  in.  9 

Ans.   70-^  sq.ft. 

10.  The  base  of  a  cylinder  is  24:  sq.  in,,  and  its  altitude 
is  2  ft.  9  in. ;  what  is  its  content  ?  Ans.   Q%  cii.  ft. 

11.  What  is  the  content  of  a  room  whose  length  is  l^ft. 


MENSURATION.  305 

6  in.,  whose  breadth  is  12 ft  4:  in.,  and  whose  height  is 
10ft.  2  in.?  Ans.  2,319ff  cu.  ft. 

12.  What  is  the  area  of  a  floor  whose  length  is  25  ft. 
3  in.,  and  whose  breadth  is  20  ft.  6  in.  f 

Ans.  611!^ sq.ft. 

13.  What  is  the  content  of  a  box  Ift.  6  in.  long, 
1ft.  3  in.  wide,  and  1ft.  1  in.  deep  ?         Ans.  2-^cu.ft. 

14.  What  is  the  content  of  a  cube,  each  edge  of  which 
is  3 ft.  4 in.  9  Ans.  37^  cu.  ft. 

REVIE^A^     aUESTIONS. 

(271.)  What  is  mensuration?  (272.)  What  is  a  polygon? 
Sides?  Vertices?  (273.)  What  is  the  area  of  a  polygon? 
(274.)  Define  a  triangle.  Its  base.  Its  altitude.  (275.)  What 
is  a  parallelogram?  A  rectangle?  (276.)  What  is  a  trapezoid? 
Its  lower  and  its  upper  bases?  Its  altitude?  (277.)  What  is  a 
prism ?  Its  bases  ?  Its  lateral  faces ?  Its  altitude?  (278.)  What 
is  a  pyramid?  Its  base?  Its  lateral  faces?  Its  altitude?  What 
is  a  frustum  of  a  pyramid?  (279.)  What  is  a  cylinder?  Its 
bases?  Its  convex  surface?  Its  altitude?  (280.)  What  is  a 
cone  ?  Its'  base  ?  Its  convex  surface  ?  Its  vertex  ?  Its  altitude  ? 
What  is  a  frustum  of  a  cone?  (281.)  What  is  a  sphere?  Its 
centre?  A  diameter?  A  radius?  (282.)  What  is  the  relation 
between  the  sides  of  a  right-angled  triangle?  (283.)  What  is 
the  length  of  a  circumference?  (284.)  The  area  of  a  triangle? 
(285.)  Of  a  parallelogram ?  (286.)  Of  a  trapezoid?  (287.)  Of 
a  circle  ?  (288.)  Of  the  surface  of  a  sphere  ?  (289.)  What  is  the 
content  of  a  parallelopipedon,  prism,  or  cylinder?  (290.)  Of  a 
pyramid  or  cone  ?  (291.)  Of  a  sphere?  (292.)  What  is  a  board 
foot?  How  do  you  find  the  number  of  board  feet  in  a  board 
or  plank  ?  (293.)  How  is  timber  measured  ?  (294.)  What  is 
the  method  of  duodecimals  ?  Define  primes,  seconds,  and  thirds. 
(295.)  Give  the  rule  for  multiplication. 


1.  Find  the  product  of  the  sum  and 
difference  of  25  and  16. 

2.  Divide  the  difference  between  1296  and 
441  by  the  sum  of  36  and  21. 

3.  What  are  the  prime  factors  of  9,800  ? 

4.  Eesolve  3,990  into  prime  factors  ? 

5.  Find  the  g,  c.  d,  of  2,290  and  458. 

6.  What  is  the  g,  c.  d.  of  1,435,  1,085,  and  2,135  ? 

7.  Find  the  I  c.  m.  of  15, 18, 24,  40,  and  50. 

8.  What  is  the  I  c.  m.  of  508  and  889  ? 

9.  Add  I,  J  off,  and  M. 

10.  Subtract  ^  of  4^  from  f  of  9^. 

11.  Multiply  \  of  2-J  by  i  of  3f 

12.  Divide  2f  by  If 

13.  Multiply  3.31  +  4.06  by  8.13—3.43. 

14.  Divide  3.8  +  2.05  by  8.6-3f. 

15.  A  man  bought  a  horse  and  carriage ;  the  horse  cost 
I  as  much  as  the  carriage,  and  both  together  cost  $640 ; 
what  was  the  cost  of  the  horse  ? 

16.  At  a  certain  election  the  successful  candidate  had  a 
majority  of  120,  which  was  -}■  of  all  the  votes  cast ;  how 
many  votes  did  the  opposing  candidate  receive  ? 


MISCELLANEOUS    EXAMPLES.  307 

17.  Divide  1357  among  A.,  B.,  and  C,  so  that  B.  shall 
receive  2^  times  as  much  as  A.,  and  C.  as  much  as  A.  and 
B.  together. 

18.  A.  can  do  a  piece  of  work  in  3  days,  B.  can  do  it  in 
4  days,  and  C.  can  do  it  in  5  days ;  how  long  will  it  take 
them  to  do  it  together  ? 

19.  How  many  bushels  of  oats  can  be  raised  on  4J  acres, 
if  each  acre  produces  47  iti.  3  pks.  f 

20.  How  many  bushels  of  wheat  at  11.75  per  bushel  will 
it  take  to  pay  for  3  civt,  of  pork  at  17  per  civt.  f 

21.  A  grocer  mixes  120  lis.  of  sugar  at  10  cts.  a  pound, 
140  lbs,  at  12  cts.,  and  60  lbs.  at  14  cts. ;  at  what  rate  must 
he  sell  it  to  clear  20^  on  its  cost  ? 

22.  Divide  11,000  among  3  persons  in  the  proportion  of 
5,  7,  and  8. 

23.  Bought  a  horse  for  $312  and  sold  him  at  a  loss  of 
121%;  ^^^^  <ii<i  I  receive? 

24.  A  man  travels  100  miles  by  rail  and  100  miles  l^y 
stage  ;  his  average  rate  of  travel  is  16  miles  per  hour 
and  the  rate  of  the  train  is  40  miles  per  hour ;  what  is  the 
rate  of  the  stage  ? 

25.  A  tank  11  ft.  deep,  12  ft.  long,  and  9  ft.  broad  is  full 
of  water ;  what  is  the  weight  of  the  water,  if  each  cubic 
foot  weighs  62J  lbs.  9 

26.  Three  men  can  do  a  piece  of  work  in  6  days ;  the 
first  can  do  it  in  15  days,  and  the  second  can  do  it  in  12 
days ;  how  long  would  it  take  the  third  to  do  it  ? 

27.  A  man  bequeathed  $37,000  to  his  family ;  he  gave 
J  to  his  wife,  ^  to  his  son,  and  divided  the  rest  equally  be- 
tween 5  daughters ;  how  much  did  each  daughter  receive  ? 


308  MISCELLANEOUS    EXAMPLES. 

28.  A  merchant  purchased  cloth  to  the  amount  of 
$1,250  and  silk  goods  to  the  amount  of  1900  ;  he  sold  the 
former  at  a  profit  of  20^  and  the  latter  at  a  loss  of  10^; 
how  much  did  he  gain  by  the  operation  ? 

29.  A.,  B.,  and  C.  enter  into  partnership ;  A.  puts  in  ^ 
of  the  capital,  B.  puts  in  J  of  the  capital,  and  C.  puts  in 
the  rest ;  at  the  end  of  the  year  their  profit  amounts  to 
$10,440 ;  what  is  C.'s  share  of  the  profit  ? 

30.  I  bought  a  lot  for  $700  and  after  holding  it  for  1 
year,  sold  it  at  an  advance  of  20^  on  the  cost  and  interest 
at  7%;  what  did  I  get? 

31.  A  merchant  bought  cloth  to  the  amount  of  $1,500 
and  sold  it  again  for  $1,770;  what  was  his  gain  %? 

32.  A  merchant  bought  goods  for  $3,000  cash,  and  sold 
them  again  for  $3,810  on  a  credit  of  four  months;  what 
was  his  gain  ^  in  addition  to  interest  on  his  money  at  the 
rate  of  Q%  ? 

33.  A  merchant  sells  cloth  at  $3.12^  per  yard  and  clears 
25^;  what  ^  would  he  clear  if  he  were  to  sell  it  for  $3.50 
per  yard  ? 

34.  Two  couriers  start  together  from  the  same  point  and 
travel  in  the  same  direction  ,*  the  first  travels  23  miles  in 
3 J  hours  and  the  second  travels  11  miles  in  2 J  hours; 
how  far  apart  are  they  at  the  end  of  31 J  hours  ? 

35.  A  laborer  spent  25^  of  his  week's  wages  for  flour 
and  had  $11.25  left ;  what  did  he  receive  per  week  ? 

36.  John  Churchill's  farm  is  composed  of  36  acres  of 
pasture  land,  22  of  meadow  land,  18  of  plough  land,  and 
20  of  woodland;  supposing  it  to  be  rectangular  in  shape 
and  128  rods  long,  what  is  its  width  ? 


MISCELLANEOUS    EXAMPLES.  309 

37.  If  30  bushels  of  wheat  cost  $67.50,  how  much  can 
be  bought  for  $438.75  ? 

38.  If  12  men  can  build  a  wall  in  20  days,  how  many 
men  will  be  required  to  do  the  same  work  in  8  days  ? 

39.  If  $100  gain  $6  in  9  months,  what  principal  will 
gain  $11  in  5  months  ? 

40.  A  certain  quantity  of  hay  will  feed  963  sheep  for 
7  weeks ;  how  many  must  be  turned  away  that  it  may  feed 
the  remainder  for  9  weeks  ? 

41.  The  third  part  of  an  army  were  killed,  the  fourth 
part  were  taken  prisoners,  and  there  remained  10,800; 
how  many  men  did  the  army  contain  ? 

42.  Thomas  sold  600  pineapples  at  16f  ds.  each,  and 
received  as  much  as  Henry  did  for  a  number  of  melons  at 
40  ds. ;  how  many  melons  did  Henry  sell  ? 

43.  A  flag-staff  stands  \  of  its  length  in  the  ground, 
12  feet  in  the  water,  and  f  of  its  length  in  the  air ;  what 
is  the  length  of  the  staff? 

44.  S.,  J.,  and  B.  enter  into  partnership ;  S.  puts  in 
$5,600,  J.  $4,900,  and  B.  $3,500 ;  if  they  gain  $1,650,  how 
much  will  each  gain  ? 

45.  A  shipper  insures  $2,500  worth  of  oats  at  4^;  for 
what  must  he  insure  to  receive  the  value  of  his  oats  and 
the  cost  of  insurance  in  case  of  loss  ? 

46.  A  merchant  insures  $4,000  worth  of  silk  at  2^^; 
what  must  be  the  face  of  his  insurance  that  he  may  lose 
nothing  in  case  of  its  destruction  by  fire  ? 

47.  A  merchant  bought  several  bales  of  cloth,  each  con- 
taining IdZ^yds.,  at  the  rate  of  12  yds.  for  $11,  and  sold 


310  MISCELLANEOUS     EXAMPLES. 

it  at  the  rate  of  8  yds.  for  $7,  losing  $100  by  the  transac- 
tion ;  how  many  bales  did  he  buy  ? 

48.  A.  owes  B.  $2,500,  payable  in  4  months,  but  at  the 
end  of  3  months  he  pays  him  $1,500 ;  how  long  after  this 
payment  before  the  balance  is  equitably  due  ? 

49.  What  is  the  bank  discount  on  an  accommodation 
note  of  $3,000  for  60  days  at  7^  ? 

50.  A  lady  wishes  to  carpet  a  floor  lb  ft,  din,  wide  and 
22 ft.  6  in.  long,  with  carpeting  f  yd.  wide ;  if  the  carpet- 
ing is  worth  $2.50  per  yd.  how  much  will  it  cost  ? 

51.  Three  men  hire  a  pasture  for  1  year  and  pay  $45  for 
its  use ;  the  first  puts  in  100  head  of  cattle,  the  second 
puts  in  150  head,  and  the  third  puts  in  50  head :  what 
must  each  pay  ? 

52.  How  many  board  feet  in  250  planks,  each  14/i^.  long, 
1 6  in.  wide,  and  2 J  in.  thick  ? 

53.  A  pile  of  wood  is  3^  feet  wide,  5 J  feet  high,  and 
147  feet  long ;  how  many  cords  does  it  contain  ? 

54.  A.,  B.,  and  C.  undertake  a  job  for  $400 ;  A.  furnishes 
4  men  for  8  (^ays,  B.  6  men  for  7  days,  and  C  13  men 
for  2  days:  what  share  of  the  money  ought  each  to 
receive  ? 

55.  A  merchant  bought  a  piece  of  merino  containing 
32  yards  for  $25.60,  and  then  marked  it  so  that  he  could 
fall  4:%  on  the  asking  price  and  still  make  20^  on  its  cost ; 
what  did  he  mark  it  per  yard  ? 

56.  A  house  is  40  feet  from  the  ground  to  the  eaves,  and 
a  ladder  is  placed  with  its  foot  30  feet  from  the  house ; 
how  long  must  it  be  to  reach  the  eaves  ? 

57.  A  man  buys  f  of  a  piece  of  property  and  sells  20^^ 


MISCELLANEOUS     EXAMPLES.  '  311 

of  his  share  for  $5,000,  clearing  25%  on  its  cost;  what 
was  the  original  value  of  the  whole  property  ? 

58.  K  wheat  at  Ss.  M.  per  bushel  gives  a  profit  of  10^, 
how  much  will  it  give  if  sold  at  95.  ^d.  per  bushel? 

59.  Of  the  trees  in  an  orchard  ^  are  apple  trees,  ^  peach 
trees,  -J-  plum  trees,  and  the  remaining  200  are  cherry 
trees ;  how  many  trees  in  the  orchard  ? 

60.  If  a  quantity  of  bread  will  last  1,500  men  for  12 
weeks  at  the  rate  of  20  oz.  per  day  for  each  man,  how  long 
will  the  same  bread  last  2,500  men  at  the  rate  of  16  oz.  per 
day  for  each  man  ? 

61.  A  path  3  feet  wide  runs  around  a  rectangular  yard 
whose  length  is  105  yds,  and  whose  breadth  is  ^5  yds.;  if 
the  outer  edge  of  the  walk  is  4  feet  from  the  wall,  how 
many  square  feet  will  it  contain  ? 

62.  A  man  leaves  $38,000  to  be  divided  among  3  sons 
and  3  daughters ;  each  son  is  to  receive  33-J^^  more  than 
the  eldest  daughter  and  each  of  the  younger  daughters  is  to 
receive  33-J^  less  than  the  eldest :  what  is  the  share  of  each? 

63.  If  a  clock  beats  31  times  in  30  seconds,  how  many 
times  will  it  beat  in  3  da.  5  hrs.  ^min.9 

64.  A  man  agrees  to  execute  a  contract  in  60  days  and 
places  30  men  on  the  work ;  at  the  end  of  48  days  the  job 
is  but  half  completed :  how  many  men  must  he  employ 
the  rest  of  the  time  to  fulfill  his  contract  ? 

65.  A  man  sells  eggs ;  to  the  first  person  he  sells  half 
his  stock  and  one  more,  to  the  second  person  he  sells  half 
of  what  remains  and  one  more,  and  to  the  third  person  he 
sells  half  of  what  remains  and  one  more,  when  he  has  none 
left :  how  many  had  he  at  first  ? 


312  MISCELLANEOUS     EXAMPLES. 

66.  At  what  time  between  5  and  6  o'clock  are  the  hour 
and  minute  hands  of  a  clock  together  ? 

67.  If  30  men  require  40  days  to  do  a  piece  of  work, 
how  many  men  will  be  required  to  do  5  times  as  much 
work  in  one  fifth  of  the  time  ? 

68.  A  gentleman  being  asked  his  age,  replied,  if  you 
add  to  it  its  half,  its  third,  and  three  times  three,  the  sum 
will  be  130  ;  what  was  his  age  ? 

69.  A.  and  B.  together  can  do  a  piece  of  work  in  18 
days,  but  with  the  assistance  of  C.  they  can  do  it  in  11 
days ;  in  what  time  could  C.  do  it  by  himself? 

70.  A.  starts  from  Bantam  at  97i.  lor/z.  A.  m.  and  travels 
toward  Norfolk  at  the  rate  of  4  miles  per  hour ;  B.  starts 
from  Norfolk  at  9  A.  30  w.  A.  M.  and  travels  toward 
Bantam  at  the  rate  of  3^  miles  per  hour ;  the  distance 
between  the  two  places  is  21  miles :  at  what  time  will  they 
meet? 

71.  If  3,000  copies  of  a  book  of  11  sheets  require  66 
reams  of  paper,  how  much  paper  will  5,000  copies  of  a 
book  of  12|  sheets  require  ? 

72.  If  24  men  can  reap  76  acres  in  6  days,  how  long 
will  it  take  18  men  to  reap  114  acres  ? 

73.  If  10  men  can  blast  30  cu.  yds.  of  rock  in  8  days, 
how  many  cu.  yds.  can  20  men  blast  in  10  days  ? 

74.  If  7  men  can  mow  84  acres  in  12  days  of  8J  hours 
each,  in  how  many  days  of  7|  hours  each  can  20  men  mow 
208  acres  ? 

75.  How  many  acres  of  land,  at  $150  per  acre,  must  be 
given  for  750  hhls.  of  flour,  at  14.60  per  barrel  ? 

76.  How   many  kilogrammes  of   butter,   at   50  cts,   a 


MISCELLAKEOUS     EXAMPLES.  Bl3 

hilog.,  must  be  given  for  1  meters  of  cloth,  at  $4.50  per 
meter  ? 

77.  How  much  cloth,  at  20//*.  per  meter,  must  be  given 
for  a  watch  worth  315.20/r..^ 

78.  If  3  trees  furnish  8.1  steres  of  timber,  what  will  62 
trees  of  the  same  kind  produce  ? 

79.  If  5  children  eat  21  liectog.  of  cake,  how  much  will 
93  children  eat  ? 

80.  Divide  55/r.  35  c.  among  7  men  and  6  women, 
giving  to  each  man  3  times  as  much  as  to  each  woman. 

81.  A.  and  B.  commence  trade ;  A.  puts  in  $350  for 
8  months,  B.  $600  for  7  months,  and  they  make  $700 ;  to 
what  part  of  the  gain  is  each  entitled  ? 

82.  W.  and  B.  engage  in  business ;  W.  puts  in  $18,000 
for  17  mo.  and  B.  puts  in  $24,000  for  6  mo. ;  while  in 
business  they  lose  $6,500 ;  what  loss  must  each  bear  ? 

83.  On  the  1st  of  January,  A.  commenced  business  with 
a  capital  of  $17,000;  on  the  1st  of  April  B.  entered  the 
business,  advancing  $12,000  capital;  and  on  the  1st  of 
July  0.  was  admitted  and  advanced  $16,000;  at  the  end 
of  the  year  the  firm  had  gained  $8,160.  How  much  of 
the  gain  ought  each  to  receive  ? 

84.  A.,  B.,  and  0.  have  business  transactions  together 
whereby  they  gain  $18,049.60;  A.  furnished  $22,000  for 
12  months,  B.  $18,600  for  10  months,  and  C.  $30,000  for 
7  months  ;  to  what  part  of  the  gain  is  each  entitled  ? 

85.  A  grocer  has  two  kinds  of  tea ;  the  better  kind  is 
worth  $1.20  a  pound  and  the  poorer  kind  is  worth  75  cts. 
a  pound;  in  what  proportion  must  he  mix  them  that  tlie 
mixture  may  be  worth  $1  a  pound  ? 


314  MISCELLANEOUS    EXAMPLES. 

86.  A.  and  B.  together  can  do  a  job  in  7  days,  but  it 
would  take  A.  alone  12  days  to  do  it ;  how  long  would  it 
take  B.  alone  to  do  it  ? 

87.  A  father  left  $10,000  to  his  two  sons,  aged  respec- 
tively 14  and  18,  to  be  divided  between  them  so  that  the 
shares  at  simple  interest  at  6%  should  be  the  same  when 
each  was  21  years  old ;  what  was  the  share  of  each  ? 

88.  A.  and  B.  commenced  business  with  equal  sums  of 
money  ;  at  the  end  of  one  year  A.  had  gained  a  sum  equal 
to  J  of  his  original  capital  and  B.  had  lost  15,000 ;  A.  then 
had  twice  as  much  as  B.;  what  was  the  original  capital  of 
each? 

89.  A  man  divided  his  estate  into  three  equal  parts, 
giving  to  his  wife  1200  more  than  -J-  of  the  whole ;  to  his 
son  1400  more  than  J  of  the  whole,  and  to  his  daughter 
1600  more  than  ■}-  of  the  whole ;  what  was  the  value  of  the 
estate  ? 

90.  A  manufacturer  employed  men,  women,  and  boys  ; 
he  had  3  women  to  every  2  men,  and  3  boys  to  every 
2  women ;  to  the  men  he  paid  $1,  to  the  women  50  cts.f 
and  to  the  boys  25  cts.  a  day ;  at  the  end  of  6  days  he  paid 
them  all  1222.    How  many  men  did  he  employ  ? 

91.  The  head  of  a  fish  was  9  inches  long;  its  tail  was  as 
long  as  its  head  and  half  of  its  body ;  and  its  body  was  as 
long  as  its  head  and  tail  together ;  how  long  was  the  fish  ? 

92.  A  father  distributed  to  his  three  sons  A.,  B.,  and  C, 
a  sum  of  money,  giving  to  A.  |4  as  often  as  to  B.  $3,  and 
to  C.  $5  as  often  as  to  B.  $6 ;  if  A.'s  share  is  $5,000,  what 
does  each  of  the  others  get  ? 

93.  A  prize  of  $945  is  divided  amongst  a  captain,  4  men. 


MISCELLANEOUS    EXAMPLES.  316 

and  1  boy;  the  captain  has  IJ  shares,  each  man  1  share, 
and  the  boy  -J-  of  a  share ;  what  does  each  receive  ? 

94.  A  bankrupt's  assets  amounted  to  $4,000 ;  to  A.  he 
owed  i  of  the  assets,  to  B.  ^  of  the  assets,  to  0.  |  of  the 
assets,  and  to  D.  J  of  the  assets.  The  entire  assets  being 
divided  among  these  4,  what  did  A.  receive  ? 

95.  How  many  posts  7  ft.  apart  will  be  required  in 
fencing  a  rectangular  lot  containing  70,756  sq.  ft,  the 
length  of  the  lot  being  4  times  its  breadth  ? 

96.  Divide  $1,000  amongst  A.,  B.,  and  C,  so  that  A. 
shall  have  $100  more  than  C,  and  B.  $95  less  than  C. 

97.  Wiiat  number  is  that  from  which  if  you  take  f  of  | 
and  to  the  remainder  add  ^  of  -^,  the  result  will  be  10  ? 

98.  A  person  asked  the  hour  of  the  day  and  was  told 
that  the  time  past  noon  was  |  of  the  time  to  midnight ; 
what  was  the  time? 

99.  When  a  man  was  married  he  was  3  times  as  old  as 
his  wife,  but  15  years  afterward  he  was  only  twice  as  old ; 
what  was  his  age  when  he  was  married  ? 

100.  A  man  going  to  market  was  met  by  another,  who 
said,  "  Good  morrow,  with  your  100  geese."  He  replied, 
"  I  have  not  a  hundred  geese,  but  if  I  had  half  as  many 
more  as  I  have,  and  2^  geese  more,  I  should  have  a  hun- 
dred ; "  how  many  geese  had  he  ? 


^NSWER^S 


^i!i^g^-v^^^.^:0=^_w^ 


-^^.-..'^^ 


Art.  20, 

Pages  21-27. 

9.  14,776. 

10.  11,803. 

11.  52,026. 

12.  11,594/15. 

13.  151,275. 

14.  7,618  2/(^5. 

15.  16,082  <ia. 

16.  1,038,957. 

17.  7,290. 
19,585. 
82,391. 
779,264  2/d;«. 
$1,624,249. 
$686,853. 
273,329. 

26.  253,693/15. 

27.  $1,041,263. 

28.  564,407. 

29.  $351,405. 

30.  206,317. 

31.  13,507. 

32.  $839. 

33.  179,580  2/cf^. 

34.  1,715,099. 

35.  1,715,369. 
35.  14,759,180. 

37.  2,159,170. 

38.  1,707,521. 

39.  6,982.126. 

40.  12,433,713. 

41.  7,921,317. 

42.  94,370,040. 

43.  $953.94. 

44.  $1,688.54. 

45.  $930.23. 

46.  $2,808.70. 

47.  $106,059.69. 


48.  $2,413,450.43. 

49.  $291,146,187.88. 

50.  $5,822.08. 

Problems. 

8.  254. 

9.  $16,985. 

10.  107,683  &?^. 

11.  687  mi. 

12.  258,928 /i{. 

13.  S,69opp. 

14.  $92,950. 

15.  3,073,134. 

16.  74,470  m». 

17.  44,437,245. 

18.  43,15iyds. 

19.  37,199. 

20.  39,351  mi. 

21.  $75,063.23. 

22.  $925.90. 

23.  $59,984.48. 

24.  $1,136.98. 

Art.  26. 

Pages  31-36. 

13.  17,571. 

14.  18,654. 

15.  23,017. 

16.  57,931. 

17.  19,238. 

18.  591,303. 

19.  666,667. 

20.  78,004. 

21.  900,497. 

22.  305,106. 

23.  37,486. 

24.  111,530. 

25.  409,095. 

26.  561,906,000. 

27.  604,918. 


28.  19,713. 

29.  403,760. 


39,990.990. 
61,303. 
78,335. 
5,333. 
28,571. 
$732,996. 
^      $206,992. 

37.  $801,965. 

38.  33,522  yJ«. 
$57,838,447. 
$4,312,956. 
3,911,106. 
178,514. 
39,499. 
4,880,874. 

^^    1,815,309. 

46.  8,600,090. 

47.  959,830 /iJ. 

48.  481,605. 

49.  530,619. 

50.  10,875. 

51.  17,138. 

52.  33,335. 

53.  $33,335. 

54.  74,835  j^. 

55.  345,153. 

56.  $34.51. 

57.  $1,781.13. 

58.  $31,186.73. 

59.  $4,349.66. 

60.  $75,785.11. 

Probleizis. 

5.  3,603. 

6.  71,837. 

7.  30,388. 

8.  38,788. 

9.  $53,806. 


ANSWERS. 


317 


10.  $36,861. 

11.  ?i;397. 

12.  $883. 

13.  $2,004. 

14.  $577. 

15.  9,417^1. 

16.  $25,600. 

17.  $9,112. 

18.  $6,740. 

19.  1897. 

20.  28,887  &?^. 

21.  Lost  $2,410. 

22.  31  mi. 

23.  123  m/. 

24.  $4,419.84. 

25.  $3,853.48. 

26.  2,331  sq.  mi. 

27.  $351.74. 

28.  1,620. 

29.  $5,318. 

30.  10,680. 

Art,  29. 

Page  40. 
9.  677,184. 

10.  203,940/^. 

11.  244,494  ;6«. 

12.  $1,665,255. 

13.  394,875. 

14.  163,636 /<. 

15.  $1,227,142. 

16.  957,504. 

17.  284,733. 

18.  1.039,660. 

19.  $717,552. 

20.  $4,233,086. 

21.  5,737,401. 

22.  5,333,328. 

Art,  32, 

Pages  JfS-U- 
8.  305,375. 

9- 
10. 
II. 
12. 

13- 
14. 

15- 
16. 


131,794. 

76,923. 

536,724. 

337,770. 

2,032.128. 

8,129,385. 

2,807,208. 


17.  5,760,757. 

18.  38,801,217. 

19.  16,179,212. 

20.  45,656,744. 

21.  183,280,678. 

22.  86,409,776. 

23.  129,414,654. 

24.  110,083,096. 

25.  28,370,748. 

26.  143,533,733. 

27.  437,557,351. 

33.  6,786.000. 

34.  27,318,000. 

35.  33,948,000. 

36.  15,400,800. 

37.  516,672,000. 

38.  1,560,793,500. 

39.  64,090,000. 

40.  16,442,400. 

41.  9,390.000. 

42.  304,741,000. 

43.  3,179,520,000. 

44.  369,369,000. 

45.  1,049,760,000. 

46.  300,000,000 

47.  3,458,280,000. 

48.  903,243,000. 

49.  183,293,000,000. 

50.  33,442,200,000. 

51.  3,199,878. 

52.  52,970,405. 

53.  13,642,498. 

54.  47,673,087. 

55.  630,063,000. 

56.  1,100.220,680. 

57.  169,589.100. 

58.  2,716,002. 

59.  120,051. 

60.  11,875,160. 


Art,  34, 

Page  45. 
125,712. 

74,508//. 
$242,235. 
3,672,672  2/<fe. 
1,034,352  ^&s. 
35,328. 

8.  759,440. 

9.  $1,820,808. 


10.  136.730. 

11.  275,184. 

Problems. 

Pages  45-48. 

5.  119,568  wm. 

6.  262,800  bbls. 

7.  W,000  rds. 

8.  $16,362,500. 

9.  1,624  &w. 

10.  4,111,200 /if. 

11.  17,920  r^«. 

12.  51,574. 

13.  153,180. 

14.  349,860  ;6». 

15. 
16. 

I?. 
18. 
19. 
20. 
21. 
22. 

23. 
24 


95,040/15. 
40,824. 
$202.50. 
$11.58. 
$11,400. 
30  mi. 
854,496. 
172  mi. 
223  mi. 
1,240  mi. 
25.  792  yds. 


Art,  40. 

Page  54. 

15.  3,090. 

16.  746. 

17.  3,367. 

18.  9,476. 

19.  11,359. 

20.  91,477. 

21.  .*91,306f. 

22.  57,799|. 

23.  45,902  Ihs. 

24.  143,071  f. 

25.  $2,379,590f. 

26.  10,290,589f  «d6, 

31.  74.074. 

32.  7,007. 

33.  619. 

34-  2,228. 

35-  12,903. 

36.  1,344. 

37.  11,451tV 

38.  10,351. 
39-  5,972. 


318 


ANSWERS. 


40.  $l,519yV 

41.  2,216tV 

42.  33,600ff. 
43-  $1,006|. 

44.  561  f. 

45.  $3.11. 

46.  $2j\. 

47.  415  lbs. 

48.  $9.21. 

49.  $8.72. 

50.  4:QSi/d8. 

51.  $4.18. 

52.  67 /if. 

^r«.  41, 

Page  57, 

5.  217. 

6.  342. 

7.  226. 

8.  lOS^ff. 

9.  102. 

10.  99. 

11.  461. 

12.  72. 

13.  284. 
14-  481AV 

15.  217. 

16.  218^VV 

17.  463^1^^ 

18.  1,003. 

19.  7,815. 

20.  96. 

21.  192  lbs. 

22.  85  yds. 

23.  l,766|f /Jf. 

24.  $356. 

25.  34. 

26.  201. 

27.  $37. 

28.  195. 

29.  2,503  mi. 

30.  203  liorses, 

31.  7,941. 

32.  3,864. 

33.  2,372ff. 

34.  133,056. 

35.  165,503. 

36.  844,101. 

37.  34,807. 

38.  1,684. 


39.  14.076. 

40.  997. 

41.  $32.83. 

42.  17,544^. 
43-  1,345. 

44.  194,877|f. 

45.  $32,528.17. 

46.  $325.49. 

47.  2,017iff. 
48    538Mf. 

49.  $70.56. 

50.  $672.90. 

51.  444. 

52.  71. 
53-  37. 

54.  869. 

55.  17. 

Art.  42. 

Pages  58-60. 

4.  86. 

5.  45,561il. 

6.  213.      " 

7.  864. 

8.  6,129^1^. 

9.  13,9223^V 

10.  245^1^. 

11.  405M|. 

13.  74,V 

14.  13AV 

15.  49AV 

16.  8AV,. 

17.  Ufi^. 

18.  9,427xV 

21.  ^U- 

22.  Hm- 

23.  lOyVl^ff. 

24.  30fH^. 

25.  30^|U. 

26.  20|f^^^. 

28.  7,630. 

29.  $370. 

30.  25. 

31.  56. 

32.  77. 

33.  825|f. 

34.  4,860. 

35.  llfH- 

36.  7fM. 

37.  34,492. 


38.  5fn. 

39.  7if. 

40.  5^. 

41.  87f|. 

Art.  43. 

Problems. 
Pages  6I-64. 

6.  $118. 

7.  448  A. 

8.  123  rows. 

9.  108. 

10.  284  bbl8. 

11.  42  yds. 

12.  92  A. 

13.  2,727  yds. 

14.  $247. 

15.  $122. 

16.  $1,998. 

17.  $10,405. 

18.  23,709. 

19.  560  mi. 

20.  $1,746. 

21.  11.749  lbs. 

22.  12,7QS  yds. 

23.  815. 

24.  140  yds. 

25.  $29,650. 

26.  $25,175. 

27.  $12. 

28.  36  hrs. 

29.  2,074  and  8,296. 

30.  312. 

31.  1000. 

32.  dau.  $12,923  ; 

each  son  $13,763. 

33.  $185.25. 

34.  102  da. 

35.  132. 

36.  48  bbls. 

37.  114  mi. 

38.  $23. 

39.  $86^,  at  $42  an  A 

40.  66  horses. 

Art.  48. 

Pages  67-68. 

8.  2.  3.  67. 

9.  3.  7.  79. 
10.  3.  5.  7.  47. 


ANSWERS. 


319 


II. 

2.  3.  7.  37. 

12. 

3.  5.  73. 

13- 

2.  3.  5.  7.  11. 

14. 

2.  3.  5.  7.  13. 

15. 

2.  2.  2.  3.  5. 

11. 13. 

16. 

2.  2.  3.  7.  7. 

13. 

17- 

2.  2.  2.  2.  3 

3.13. 

18. 

2.  3.  131. 

19. 

2.  2.  2.  2.  2 

2.  7.  7. 

20. 

2.  3.  5.  31. 

21. 

3.  5.  97. 

22. 

5.  11.  67. 

23- 

2.  11.  79. 

24. 

3.  5.  7.  31. 

25- 

7.  11.  13. 

26. 

5.  13. 17. 

27. 

3.  7.  151. 

28. 

13.  13.  17. 

Art.  51, 

Page  70. 

7.  16. 

8.  93. 

9.  52. 

10.  4if. 

11.  400. 

12.  57. 

13.  31. 

14.  14/0. 

15.  5tV 

16.  263^. 

17.  5. 

18.  1. 

19.  lif. 

20.  11. 

21.  153V 

22.  3. 

23.  2. 

24.  6i. 

Problems. 

5.  2^Ct8. 

6.  64  6m. 

7.  $2. 

8.  6  fir'kins. 
g.  58  boxes. 

10.  27  lbs. 


Art,  54:, 

Page  74- 

2.  4. 

3.  45. 

4.  63. 

5.  15. 

6.  66. 

7.  210. 

8.  64. 

9.  108. 

10.  81. 

11.  42. 

12.  630. 

13.  267. 

14.  396. 

15.  13 

Art.  56, 

Pages  75-76. 

1.  267. 

2.  396. 

3.  414. 

4.  533. 

5.  84. 

6.  33. 

7.  23. 

8.  87. 

9.  630.    • 

10.  267. 

11.  396. 

12.  72. 

13.  135. 

14.  72. 

15.  23. 

16.  252. 

17.  3. 

18.  12. 

19.  8. 

20.  4. 

21.  37. 

22.  108. 

23.  73. 

24.  76. 

25.  55. 

26.  83. 

Art.  59, 

Page  78. 
I.  24. 
•  2.  2.520. 


3.  1,008. 

4.  5,048. 

5.  13,860. 

6.  2,520. 

7.  540. 

8.  420. 
720. 
1,176. 
16.800. 
3,528. 
44,100. 
14,700. 
468. 
5,070. 
3,400. 
4,275. 
13,475. 


20.  1,512. 

Problems. 

Page  79. 

1.  9  or  $9. 

2.  180/i5. 

3.  45  bu. 

4.  120  ft. 

5.  13. 

6.  30  qts. 

Art.  67. 

Page  85. 
I.  Y- 

3.  ¥• 

4.  V-. 

5.  W- 

6.  ^^K 

7.  Hi^. 

8.  ^^K 

9.  ^V¥-^. 

10.   4-VV-^. 

Art,  68, 

Page  86. 

207 
347 

ss-- 

114 

w. 

432. 


317221 


10.  20H|. 

II.  ^^. 

12.  25. 
1 13.  3H|. 


14 

4. 

15 

22|f. 

i6 

pis 

17 

18 

iff.* 

19 

33tf|. 

20 

393/A. 

21. 

108||. 

22 

18Hf. 

23 

Ififff. 

24. 

225/A. 

25. 

1453:«,V 

26. 

281. 

27. 

434. 

28. 

4^TT. 

29. 
30. 

If- 

^?'/?.  70 

P«^6  <?5>. 

I. 

I. 

2. 

J" 

M- 

4. 

ii?« 

5. 

y'k* 

6. 

7. 

i. 

8. 

3 

■  y 

9- 

?• 

10. 

i^. 

II. 

U- 

12. 

If. 

13. 

14. 

M* 

15. 

?¥• 

16. 
17. 

!|: 

18. 

i. 

19. 

|. 

20. 
21. 

:|: 

22. 

T' 

23- 

i« 

24. 

25- 

■Jg. 

26. 

tf* 

27. 

It* 

28. 

"A"* 

29' 

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ANSWEBS, 

32. 

f 

33. 

34. 

'• 

55. 

ff. 

36. 

37. 

1*. 

38. 

A. 

39. 

H- 

^r«. 

71. 

Page  90. 

2. 
3. 

ft; 

4. 

nnr* 

5. 

6. 

W' 

7. 

AV 

8. 

*• 

9'M 


Art.  72. 
Page  91. 

15     30     18 
3Ty»   30»   3^* 

72         81  48 

TTF8r>  TTJ¥'  TTI¥) 
168  64  140 
^2Tj  ^^¥»   ¥^¥» 

495  780  1001 
"JTT^J  ^Tl?»  ^T¥¥' 
80482      7315     12005 

1564     5681      516>8 
TT^7'    7¥¥7»   7¥^9- 
19720    17340     3451 
3  3¥^¥>33  5^¥>¥¥5^¥' 
1716      8993      3861 
TYT^^J  57T7»  ¥TT¥- 
416      234     588 
11T»  ¥^¥'  ¥^T- 
2116      2898      4554 
?5^^»   ^^¥1?>    7^¥^- 

Page  92. 

24      30      72      18 
?!¥?  ¥T»  FT»  ¥¥> 
164       90       176       54 
!¥■?»  T¥¥>  T7¥»  TITF- 
234      198      390       66 
¥55>  FF5»  ¥55>  5BF* 
108       90       140 
T¥TJ>  T'SIS'  T??TF- 
840      252      264 

TJ¥»  ¥S^«^»  arj^« 

270      63      88 
680         330        1848 

H.  ii  ^. 
If  >  /?.  If. 

3        19      85 
¥¥'  ¥Tr>    ¥IT' 

MH,  A*^,  ^V^. 
|Vt,  ifl,  A\- 

TITOT'  3  0011  TTTffT* 


^°-  T^¥T»  1%¥4.  T-gi?- 

IQ  f4-l     ^2  0      886     42  3 

^9*  f¥3^>  ¥¥7»  TFT*  f  tl 

2n  JjaO        7  8         2  8         5  9 

2T      ^5-     i-O      4  5      4  8      5  0 

6/1  ¥Tr>  ^17»  ¥11'  try- 

22      _6  6         6  8         6  6         4  9 
22-    ^T»   ^3T>  fST*   131^ 

Mf. 

2-5  4-8  5  1  328  78  1 
^>  f  6>  6^1  SW-y  -¥¥-• 
OA      J6  7.8       5  7       3  0  5       6  4 

■''J'    T6¥>   Te?'    16#»   T6  8" 
26      128  1      195       9  8     124 

27.  Z^^>^^^>      3-00., 

28.  ^v-,lil!iif,AV 

29.  Wt\JA¥-,  m. 

30.  -VW->^Vt¥-,  W/, 

Mi. 

-4^f.  75. 


Pa^e  5<^. 


1  23 

ItVo- 
1  1 1 

117 

15|f. 


II.  2ft^. 


~36t 

12.  2i|. 

13.  2fH. 

14.  2A. 

19.  22^^. 

20.  672/^. 

21.  624|. 

22.  106/y. 

23.  1.357x«^. 

24.  134/^. 

25.  llii§. 

26.  20AV 

27.  15, 

28.  1.^ 

30.  14f^. 

31.  23f|. 

^3:  ISr 


ANSWERS. 


321 


34.  34H. 

27 

^%mi. 

35.  36||. 

28 

$215^. 

36.  9^^. 

29 

48|  T. 

37.  lO^m. 

30 

79f  ^. 

38.  mi 

31 

82f  1  in. 

39.  19i^. 

32 

milrds. 

4o.  lO/A. 

33 

$124|-;. 

41.  mn- 

34 

201^  rds. 

42.  07,%V  Ws. 

35 

454i  hu. 

43.  486J-I. 

Problems. 

44-  3253^,=,,  yds. 

Pages  08-99. 

Problems. 

4- 

%^r^,yds. 

Page  96. 

5- 

94i|  A. 

3.  m^mi. 

6. 

42ff^6«^. 

4.  57,%^Ar«. 

7. 

94f  .V  tons. 

5-  .$394,641. 

8. 

241  yds. 

6.  $43f. 

9- 

$39/0. 

7.  123|f^^&*. 

10. 

$3^-. 

8.  5-||fA  ^o;^s. 

II. 

'^I'^lyds. 

9-  IBS/ffVy^*. 

12. 

33.1  yni. 

10.  im\A. 

13. 

16 1  gcd. 

Art.  77. 
Page  97. 

14. 
15- 
16. 

54:lbs. 

$11H. 

gain^li^. 

I-  A. 

17. 

9,r,  m^-. 

2.  iV- 

18. 

mi^  yds. 

3.  4^1,. 

^'  Ik 

5.  4,3^. 

6.  3A. 

7.  IIA. 

19. 

lost  .$131  x'^. 

Art.  79. 

Pages  101, 102. 

6. 

__7_ 

8.  65/j. 

7. 

7|f. 

9.  31il. 

8. 

9|f. 

IO.  ^. 

9. 

65^. 

ir.  52f. 

10. 

45H4. 

12.  84i§. 

II. 

34. 

13.  8A. 

12. 

66. 

14.  2H. 

13. 

70^. 

15.  5.V 

16.  749^?A. 

14. 
15- 

121-1. 

17.  $5^V- 

16. 

l,648i. 

18.  m^ft. 

17. 

10/7. 

19-  ^m- 

18. 

M 

20. 11- ^d«. 

19. 

30^^. 

21.  20|^  pds. 

20. 

§. 

22.  $94 fV. 

21. 

2fV. 

23.  |12fVo. 

22. 

21^. 

24.  51H|. 

23- 

13^ 

25.  582|fl. 

24. 

6. 

26.  llOpw. 

25- 

4,V.. 

11 


26.  9,333ft^. 

>7>7        355 

27.  T¥T6. 

28/ ll. 

29-  m- 

30.  lf||. 

31.  24,450f. 

32.  tV 

33.  561f^. 

34.  1.841  ,V 

35.  31,79011. 

36.  8,3471^. 

37.  6,199t|. 

38.  56,455|t. 

39.  99,151*. 

40.  19,166^1. 
Problems. 

102. 10  J. 


5.  49^%  ^*' 

7.  13ff  hu. 

8.  $43f. 

9.  |17,llli|i. 
10.  gain  $13. 


13.  333f  ^. 

14.  28|  mi. 

15.  66^  yrs. 

16.  $§: 

17.  297|  2^<?s. 

18.  185|  mi. 

Art.  81. 

Pages  105, 106. 

3.  f 

4.  ^h- 

5.  lA- 

6.  If. 

>^     188 


20 

21.  ^M 

22.  224 


ll — 


r5¥* 


23.  1 

24.  5t=V 

25.  M. 

26.  8||. 

27.  sHi)- 

28.  M- 

29.  3^1^. 

30.  1//^. 

31.  IN,' 

32.  34|ff 
i  33.  ^eV 


35.  Hi 

36.  14 

37.  31 


17.  5^Ty. 

18.  16. 

19.  39Mf. 


38. 

6^¥^. 

39. 

9tV. 

40 

^vv. 

Problems. 

Pages  107, 108. 

7- 

10t\. 

8. 

5M. 

9. 
10. 

|!04. 

II. 

419^1  ;&5.; 

12. 

131A  lbs. 

13- 

li  e?«. 

14. 

d&l    39 

2m  lbs. 

15. 

16. 

45  da. 

17. 

7^  mi.; 

42  mi. 

18. 

$49,200. 

19. 

11  da. 

20. 

6^V  ^' 

21. 

$2i 

22. 

12  mi 

23. 

$23,^A- 

24. 

973  men. 

25- 

$30,000. 

Art.  82, 

Page  109. 

3.  9,850. 

4.  93,100. 


322 


AKSWERS. 


5.  203,075. 

6.  255,025. 

7.  109,650. 

8.  394. 

9.  3,724. 

10.  3,501. 

11.  1,854. 

12.  6,961. 

13.  1,012.3. 

14.  11,425. 

15.  $60,425. 

16.  $12,500. 

17.  948,400 /if. 

18.  2S4,rmyds. 

19.  $545,625. 

20.  426,437iy<fo, 

21.  810. 

22.  914. 

23.  ISO  yds. 

24.  405. 

25.  $873. 

26.  261. 

27.  $18. 

28.  soft. 

Art.  89. 

115,  IIG. 


1.  .75. 

2.  .9375. 


.421875. 
.952. 
.7616. 
.27168. 
.508. 
.048. 
9.  .0024. 

10.  19.875. 

11.  24.52. 

12.  11.1171875. 

13.  110.032. 

14.  21.00224. 

15.  4.093125. 

Art,  90. 

Page  117. 

1.  .1071. 

2.  .5135. 

3.  .7143. 

4.  .1905. 

5.  3.8235. 

6.  4.9091. 


9- 
10. 
II. 
12. 

13- 
14. 

15- 


1.625. 
1.9636. 
6.875. 
35.625. 
21.6563. 
1.0978. 
14.9333. 
13.094. 
4.7848. 
Art.  92, 
Pages  118,119. 

6.  115.652. 

7.  444.0924. 

8.  $256,017. 

9.  $144.87. 

10.  828.318  ^&«. 

11.  757.4994 ^^s, 

12.  578.1023. 

13.  247.0709. 

14.  431. 6186 /t 

15.  312.5119. 

16.  $260,889. 

17.  70.1779. 

18.  100.001  6w. 

19.  122. 

20.  81.1027. 

21.  155.3006/^., 

22.  684.2371. 

23.  $3,590.21. 

24.  231.3898  62^. 

25.  48.917  yds. 
Problems. 

Pages  119, 120. 

2.  37.495^1. 

3.  $83.92. 

4.  $47.39. 

5.  UO.lSlyds. 

6.  330.275  hu. 

7.  44.979  C. 

8.  $56,186. 97^. 

Art.  94:, 

Page  122. 
71.507. 
66.9997. 
887.8002. 
$9.77. 
1.8705 /if. 
$565,928. 
$24,998,923. 


19.  .m^yds. 
$.999. 
9.9997/^. 
5.9994  yds. 
$4,128. 
4.2197. 
6.5766. 

26.  4.3219. 

27.  4.8986. 

28.  $25,453. 

29.  76.112  Z6s. 

30.  $11,075. 

31.  127.01  lbs. 

32.  $2,761,985. 

33.  $333.13. 
Problems. 

Pages  122, 12S. 

1.  $4.05|. 

2.  25.48  yds. 

3.  $170,133. 

4.  $305. 37i. 

5.  $3,107.25. 

6.  $939.49. 

7.  $2,045.35. 

8.  $783.80. 
$963.68. 
$4,727.57. 


9- 
10. 
II. 

12.  15.684(7, 

13.  74.39 /if 


$38.29i 


14 
15 
16 

17. 


$57.05. 
123.315  Ihs. 
199.85  mi. 

^82.75. 


18.  gain  %\.\&l. 

Note.—  Answers 
are  carried  to  four 
places  in  Articles 
96-98. 

Art.  96. 

Pages  125, 126. 

3.  $641.28. 

4.  $5  3438. 

5.  0.21  6?/. 

6.  $0.0012. 

7.  lS,782.0dy ds.\ 

8.  0.0904/^. 

9.  0.0004  lbs. 

10.  42,102.3603. 

11.  1,744.3913. 


23.8894. 

386.5576. 

$8,361.32. 

$3.0088. 

$0.2475. 

$30.1534. 

$63.8406. 

>Ji0.0027. 

20.  20.4905  lbs. 

21.  84.5688. 

22.  0.3737. 

23.  412.5508. 

24.  O.nm  yds. 

25.  0.5103  rds. 

26.  87.4894. 

27.  $1,713,782. 
964.5215. 
72.5641. 
29,170.4499. 
3.3538. 
$1,485. 
81.648  ^&s. 

34.  1,594.974^<7«. 

38.  $117.33. 

39.  9,189.866  ?&s. 

40.  $4,812,975. 

41.  5dd.su  yds. 

42.  3,512.7728. 

Problems. 

Pages  12G-128. 

1.  $108.37i. 

2.  $117.56^. 
$200.59i. 
$129.68|. 
$2,675.75. 
$12.75. 
$27.18f. 
$14,114.25. 
$41,781. 
$5,699. 
$23,751. 
$192,231. 
$81.56i. 
32.045  mi. 
264.1875  mi. 
65.515  mi. 
$13,737. 
$111.58. 
$14.396^am. 
297.2538  mi. 


3. 
4. 

5- 
6. 

7- 

8. 

9- 
10. 
II. 
12. 

13- 
14. 

15- 
16. 

17- 

18. 

19 
20. 


ANSWERS. 


Art.  98. 

Pages  130, 131. 

4.  0.0025. 

5.  $1-39. 

6.  31.5434. 

7.  0.25. 

8.  87.5. 
Q.  4.75. 

10.  112.8767. 

11.  356.1111. 

12.  12.5 /i(. 

13.  12.24. 

14.  1,485.6016. 

15.  $0,167. 

16.  153.8462. 

17.  $0,061. 

18.  S2.27  yds. 

19.  0.0268. 

20.  0.916. 

21.  $0.67. 

22.  2.36. 

23.  79.52. 

24.  123.107. 

25.  7.54. 

26.  $70.55. 

30.  $16,196. 

31.  $1,439. 

32.  7.1364/15. 

33.  84.775  ytZs. 

34.  3.1034  lbs. 

Problems. 

Pages  131-133. 

5.  48.5  A. 

6.  $8.50;  $53.55. 

7.  19  0.;  3.7  C. 

8.  11.5yds.\    35.6365 

yds. 

9.  149  11)8. 

10.  S7. 5  bbls. 

11.  13.5  m*.;  15.25  m 

12.  $3. 

13.  58.8666  w^.; 
559.2333  wj. 

14.  306.675  m. 

15.  lOAhrs. 

16.  $128.50. 

17.  $1.12|;  $7.31i. 

18.  $16.08tV 

19.  $67.68. 


20.  29^ct8. 

21.  11  hrs. 

22.  72  bu. 

23.  18 cts.;  $5,624. 

24.  51  bu. 

25.  2Slbu. 

26.  215.18  m. 

27.  10.5. 

28.  7  hrs. 

29.  $27.50. 

30.  10.5(7. 

31.  $5.89. 

32.  41  T. 

33.  7  -yes^^. 

^*'*.  J  00. 

Pages  134, 135. 

3.  $62.50. 

4.  $18. 

5.  $3.75. 

6.  $23. 

7.  $3.12^. 

8.  $9.12A. 

9.  $4.25. 

10.  $23.40. 

11.  $15.66f. 

12.  $9,061. 

13.  $29.10. 

14.  $39.37|. 

15.  $23.33^. 

16.  $7. 

Art.  101. 

Pages  135,  136. 

2.     $5.17/g. 

3.  $356. 66|. 

4.  $54,461 

5.  $5,025. 

6.  $34.33i. 

7.  $35.77: 

8.  $118.14. 

9.  $262.23. 

10.  $82.87i. 

11.  $219.60. 

12.  $37.80. 

13.  $69. 

14.  $70.61. 

Art.  102. 
Page  136. 
2.  $23  59t^. 


3.  $99,533^. 

4.  $79.67^. 

5.  $407.10. 

6.  $1,029.60. 

7.  $12.96. 

8.  $63,602. 

9.  $57,833. 

10.  $66.18. 

11.  $70.18. 

12.  $8.32|-. 

13.  $22.06^. 

14.  $301,883. 

Art.  103. 
Pages  138, 139. 

1.  $102.58A. 

2.  $111.83. 

3.  $51. 

4.  $6.62. 

5.  $89.87. 

6.  $49.81. 

Art.  104. 

Page  140. 

1,  $86.28  Gr. 

2.  $247.68  Cr. 

Art,  146. 

Pages  159, 160. 

1.  4,382 /ar. 

2.  21,268  5rrs. 

3.  37,740  min. 

4.  5,iS6  yds. 

5.  10,890  sq.fi. 

6.  6615  pts. 

7.  2,739,600  sec. 

8.  30,183 /ar. 

9.  1,365  in. 

10.  175,000  5^.  E 

11.  200  cu.  ft. ;   3,200 

cu.  ft. 

12.  664:  qts. 

15.  17s.  6d. 

16.  11  OS.  5  dwts. 

17.  2  cwt.  1  qr.  12  lbs 

8oz. 

18.  4  sq.ft.  7.2  sq.  in. 

19.  £2  7s.  6d. 

20.  3  da.  12  7ir5. 15  min, 

21.  2  ^?'*.   37  w.  5  da. 

6  hrs. 


324 


ANSWEES. 


22. 

23. 
24. 

25. 

26. 

27, 
28. 
29. 

30. 

31- 
32. 

33- 
34. 
35- 
36. 

37. 

38. 
39. 

40. 
41. 


4  r.  11  cwt 

d  T.  16  cwt.  121  lbs. 

2  mi.  6  fur.  dOrds. 

7  fur.  31  rds.  Sj^ 

in. 

3  &w.  2  ^/fcs.  3  qts. 
43  2/^«.  6|  m. 

11  yds.  i ft.  Uin. 

4  M<Z«.  1  hU.  dgals 

dqts.  1.24S  pts. 
3°  42'. 
12°  9'  43".2. 

8  cwt.  2  gr«.  22  ?6s. 
175.  OU. 

4  &w.  1  pk.  31 1  ^^5. 
2  r.  11  cwt.  20  ?&s. 
3c?<z.  ISi'^rs.  36wm. 

28.8  sec. 
4:  fur.  2Srds.  Syds. 

2  ft,  2. Uin. 
89i  cts. 
'dA.   IE.  33.6  sq. 

rds. 

1  qt.  1.656  i?^«. 
^fur.  8  r(Z«.  32^e?s. 

Art.  14:8, 

Pages  162-164. 

2  lbs.  8  oz.  15  (?«^^«. 

12  ^rs. 
52  w.  1  da.  6  hrs. 
Sm  62  33  23. 
71  13  12  ^r«, 
6  7'cfe.   4  2^(?«.    2  ft. 

9  z?i. 

4  77^^'.    97  7'(?*.    Ift. 

10  m. 

Im.  126  r<Z*.  3  «/<?«. 

l/i5.  7  m. 
2  grs.  4  ?&s.  9  oz. 
2S8  gals.  2  qts.  2gi. 
£8  Us.  nd. 

2  lbs.  5  oz.  16  dwts. 

7gr. 
16m  61  63  13. 
10  c«j^.  1  gr.  11  <?2. 

3  c?a.  14  hrs.  29  m. 

35  sec. 
Swks.  20  hrs.  2Sm. 
5°  51'  58". 
18°  34'. 


4.643  m. 

10  sq.  yds.   2  sq 

114  sg^.  in. 
12  A.  Isq.ch.  1 

sq.  li. 

YE^hd. 
TT  IVli. 

■a-hrt.a.     ■ 
Ugal. 

0.714  w. 

£14.8625. 
7.871  mi. 
0.1188  r. 
0.2709  lb. 
0.2917  ^a?. 
0.58dS  yd. 
28.25  oz. 
4,475  lbs. 
d5.75  pks. 
104.75s. 
109.375  ^afe. 
263.1667  2^(?s. 
2,855'. 7. 
0.043  decam. 


.ft. 
312 


Miscellaneous. 
Pages  166, 166. 
11,400  grs. 
16  lbs.    10  oz. 

dwts.  5  grs. 
49,775  ^65. 
4,368  qts. 
24.8  mi. 
201  a 
214  (Za.  15Ar«. 

m.  35  sec. ; 
30  whs.  4  (?a.  15  Ars, 

30  m.  35  sec. 
44<Z.;  4s.  8d 
275.59m.;  511. 81?:^ 
15.367  m.;  8.1072  m, 
7.4564  mi. 
2Lim>  kilom.', 


18 


30 


12.0701  kUom. 
43.3247  g^s. 
105.668. 
119.2391 1. ; 
280.1173  I. 
25.077^.:  1.7988^. 
163.1404  ;&s. 
1,124.346  lbs. 
4.1385(7. 
4.8283  C, • 
618,0224  C2«./^. 
£0.8906. 


2.S 

277 


7526 /6s.; 
21.0812  oz. 
0.5144^A;s. 


3s.  8d  3.52 /or. 
4  fur.  2^  rds.  11  ft. 
2.64*71. 

Pa^cs  168-170. 
83  &w.  6  g^s. 
2  &?^.  2  i>A;s.  2i  qts. 
27  ^ff?s.  2  g^s.  1  pt. 
16  yds.  2  ft.  5  in. 
IdA.S  sq.  ch.  1926 

sq.  li. 
l^da.    2^  hrs.    23 

min. 
£35  15s. 

107Z&S.  4o2.  10c?«jfe. 
12  mi.  306  r^. 
£92  Id 
157  bu.  4  g^s. 
£1  8 

5.0987  da. 
£49  2s.  2|d 
2  ^&s.  6  <>2. 19  dwts. 

17  ^rs. 
lib  31  33  23  18 

grs. 
1  T.  10  cwt.  2  qrs 

4  S>s. 
39  wks.  15  hrs. 
25  mi.  253  ?*c?s.  1  yd. 

1ft.  6  in. 


ANSWERS. 


325 


24.  591  yds. 

25.  321  bu.  2  pks.  5  qts. 

26.  84.023  T. 

27.  180.5S57  lbs. 

28.  £35.2375. 

29.  89.1  da. 

Problems. 

Pages  170,  111. 

1.  103  ciot.  10  ?6«. 

2.  21  Z6s.  4  02. 

3.  297|  2^c?«. 

4.  293  A.  6  ,<?(7.  rffs. 

5.  22  yrs.  5  ??i6>. 

6.  $23,718. 

7.  91.094  ?&«. 

8.  67  A.  7  «(7.  cA. 

9.  \^^_%yds. 

10.  4  a36cw./iJ. 

Art,  152. 

Pages  173-175. 

4.  3  cwt.  1  g-r.  33  Tbs. 

5.  36  Mc?«.  49  gals. 

6.  2  2^c?5.  2  in. 

7.  4  ^4.  145  sq.  rds. 

8.  15^^  55'  57". 

9.  17tt)    81    43    33 

1 6  f/rs. 

11.  8  rds.  4  yds.  9  w. 

12.  4  r(?s.  3  yds.  1ft. 

14.  4  yrs.  0  ?«o«.  9  (Za. 

15.  58  yrs.  3  ?Aios.  34  da. 

16.  4:3  yrs.  Imo.  12  da. 

17.  8  7irA\  7  mm.  13  sec. 

18.  £1  2s.  6d. 

19.  16  gra/*.   3  qts.  3t« 

20.  2,775.S  grams. 

21.  3  <Za.  32  Ars.  43  m. 

34  sec. 

22.  3  T.  16  c«?^.  2  qrs. 

15  ^&s. 

23.  5?&s.  Idwts.  IS  grs. 

24.  684g'a^s.  2  qts.  Ipt. 

25.  15  &M.  1  p/<j.  3  g^s. 

Problems. 

Pages  175,  176. 
2.  161  A.  87  sg.  r(?s. 


3.  31i|  yds. 

4.  104  c^f-i^.  1  gr.  10  ?&s. 

5.  2  2/rs.  9  mo.  24  da. 

6.  72  yrs.  8  mo.  11  dJa, 

7.  7  yrs.  9  mo.  1  <?fl^. 

8.  283«/rs.8m(?.33d^a. 

9.  6  C.  6  C.ft.  4  cu.ft. 
12.  6°  18'  30". 

14.  90°  34'  15  ". 

15.  8°  53'  8". 

16.  37  bu.  3  pks. 

Art,  154. 

Pages  178-180. 

4.  1  T.  18  cwt.  3  qrs. 

5.  $314. 37i. 

6.  3,006. 1/r. 

7.  70.0S  Mog. 

8.  3332/<?s. 

9.  1848.75  m. 

10.  83  icks.  4  <^. 

11.  5  7ihds. 

12.  3cM?^.  1  qr.  18  lbs. 

13.  124:  yds. 

14.  £11  13s.  4f?. 

15.  113y1.  3i?. 

16.  3  da.  6  7irs.  4  min. 

17.  6  r.    14  cw^.    I  qr. 

15  ^&s. 

18.  £123  9s.  Qd. 

19.  3,317  T.  15  c^ct 

20.  87  2/^s.  1ft.  6  in. 

21.  lU^s^^a^s. 

25.  9,656  sq.  yds. 

26.  308|^  sq.  rds. 

27.  747.1111  cw.  yds. 

29.  33.632  cw. /if. 

30.  1,017.1875  cw. /if.; 
1S1.25  sq.ft. 

31.  257.375  sg./f. 

Problems. 

P^^'es  180, 181. 

1.  £501  17s.  6d. 

2.  959  mi  20  yds. 

3.  453 ?&s.  60s.  4di€ts. 

6  grs. 

4.  6310  y<?s. 

5.  16  a  5a   ..; 
3138  C2^./^. 


6.  38  oz.  14  dwts. 

7.  1  r.  17  cwt.  2  grs. 

33  lbs. 

8.  37i  m^•. 

9.  78  ^.  30  sq.  rds. 

10.  £41  3s.  Qd. 

11.  3372/<?s. 

12.  $188.31^. 

13.  255.125  yds. 

14.  17^1  5czr^s. 

15.  1,337.25  sq.  yds. 

16.  444.875  sg. /if. 

17.  27.236  2/<Zs., • 
245.125  sq.ft.; 
67.9733  sq.  yds. 

18.  7,030  lbs. 

Art,  156, 

Pages  I84, 185. 

3.  2  c?/?^,  1  qr.  18  Z&s. 

31  OS.      - 

4.  30  7:  16  CAOt.  1  qr. 

11  lbs. 

6.  3  m».  4/w?'.  1  7yZ. 

7.  3  &!/.  3  pA;s.  4  g^s. 

8.  £1  8s.  9d. 

9.  1  ?&.  3  oz.  6  £?■«?<«. 

10.  5.3  kilog. 

11.  704. 

12.  48  mi.  301  r<?s. 
14.  .09/r. 

14.  35  liters. 

15.  £4  6s  8|eZ. 

16.  49  gals.  34  g^«. 

17.  7,040. 

18.  1.42  fr. 

19.  .43  kilog. 

20.  16. 

21.  17. 

22.  3.783. 

23.  5°  13'  5". 

24.  4°  36'  19". 

25.  3.667. 

26.  400. 

27.  3,880. 

28.  4.5. 

29.  15. 

30.  1.0833. 

32.  £1  lis.  5d. 

33.  3  lbs.  8  oz.  17 

8  grs. 


326 


ANSWEKS. 


29.575. 
377. 


Problems. 

Pages  185-188. 

1.  £2  3s.;  £10  15s.; 
£27  19s. 

2.  10^  mi. ;  31^  mi. 

3.  35  times. 

4.  24  yds. 

5.  2^yds. 

6.  9  m. 

7.  $990. 

8.  ^0  panes. 

9.  $7.98. 

10.  $73.50, 

11.  $24.36. 

12.  $20.80. 

13.  $130. 

14.  720. 

15.  22. 

16.  2^.  iB.^^sq.yds 

17.  Idiots. 

18.  $147. 

19.  $l,367i. 

20.  8,400  &r^c^■s. 

21.  768. 

22.  $34,222. 

23.  $82.50. 

24.  54. 

25.  1,920. 

26.  9s.  5fc?. 

27. 1  a  22.86  cw./i5. 

28.  £1  12s.  2c?.;  £19  6s, 

29.  84.25/7'. 

30.  6  hrs.  48  min. 

31.  882. 65 /r, 

32.  7  mi.  \l^{iyds.\ 
9  mi.  1260  yds. 

33.  270  yds. 

34.  47°  15'  16  '. 
35-  86°  49'  26''. 

Art,  157. 

Page  189. 

2.  1  hr.  9  m.  37  sec. 

3.  3  Ars.  37  m.  15  st'c. 

4.  7  ^rs.  53  m.  34  sec. 

5.  1  7<.r.  27  m.  Wsec. 

6.  5  i^rs.  18  m.  41  sec. 


2  7^7•s.  34  m.  35  sec. 
46°  1'  30". 
32"  19'  30 '. 

78°  35'  45". 

139°  17'  30". 

19°  39 . 

97°  16'. 

2^rs.  Zlmin.  29.467 

sec. 
70°  57'. 

Art,  165. 

Page  193. 
63.99  lbs. 
$39. 

m.lyds. 
13.23  mi. 
$30. 
Ic^. 

3.4375  kUog. 
1221- ft. 
5.1985. 
$18. 

204.54:  Mlog. 
4.2636  lbs. 
$115,981. 
$21,058. 
£.0925. 
15f  yds. 
200  A. 
£21  12a.  Qd. 

Art.  166, 

Page  194. 


2. 

3- 
4- 
5. 
5. 
7. 
8. 

9- 
10. 
II. 
12. 

13. 
14. 

15. 
16. 

17- 
19. 
20. 


5771  lbs. 

7.  64. yds.;  'SO yds. 

8.  45. SQkilom.; 
'SS.Q4kilom. 

9.  125 hJids.;  25hhd8 
10.  40.'S2  yrs.; 

31.68  yrs. 
163.4/i(.;  140.6/if. 
74.37  mi 


59.63  m*. 

86.72  r.;  41.28  r. 
$99.70.;  $72.24. 


78.48  &«. 
66.08/r.; 
$92. 
53.1^. 


65.52  bu. 
45.92  fr. 


$5.63^. 

Syts. 

Art.  167. 

Pages  195,  196. 

43f%. 

11%. 

13i%. 

10H%. 

460^3%. 

124%. 

166f%. 

20%. 

70%. 

25%. 

74.2%. 

871%. 

34%. 


Art.   168. 

Pages  196, 197. 

$1,500. 
$420. 
15,465. 
66f  gals. 
$4,500; 
180. 
50  rds. 
60ft. 
200  sheep. 


Problems. 

Pages  197-199. 

1.  $24.25. 

2.  1,304.75. 

5.  £1,856  17s.  6d. 

4.  $4. 

5.  550. 

6.  6d0  eggs. 

7.  $50,000. 

8.  13,000. 

9.  4.8  kilom. 

10.  10.67  hectol. 

11.  $5,400.^ 

12.  $740. 

13.  53  gals. 

14.  $20,000. 

15.  31,000  men. 


ANSWERS. 


327 


i6.  $3,026.98|. 

17.  $3,540.62i. 

18.  670, 
19. 
20. 
21. 
22. 


$3,312*. 


Art.  173. 

PUges  200-201. 
2.  S168. 

$23,482.81]. 

$4,080. 

?;52.50. 

$300.93f. 

$60.37*. 

$585.75. 

1,000. 

140,000  »«. 

$43,923. 

$47.25. 

$150 I i ; 

$11,934J. 

42,997i  Ihx. 

$3,14j.~60. 

$262.50. 

ivt.  177. 

204-205. 

$24001. 
$66.25. 

$277.50. 
$468.75. 
$1,137. 


$90.75. 

$17,040. 

$14,000. 

$69. 
$232.50. 

$875. 

$618,784. 

$2,223.75. 

$713,811. 


Art.  179, 

Pages  206-200. 
2.  $1,050. 


3.  $58.50. 

5.  $20,160. 

6.  $24. 

7.  £375. 

8.  $2,500. 

9.  $35,112. 
10.  $550. 
n.  12c^s. 

12.  $50,000. 

13.  $9,000. 

14.  $325. 

15.  $330. 

16.  $23.88. 

17.  50%. 
i8.  $13,400. 

19.  $3,447.50. 

20.  $5,636.25. 

21.  25%. 

22.  $1,000. 

23.  $120. 

24.  $135. 

25.  36x\%. 

26.  36c^«. 

27.  $27,500. 

28.  $160, 

29.  $7.50. 

30.  15%. 

31.  16%. 

32.  $282. 

33.  m%' 
34. 35%. 

35.  30%,  or 
$4.20. 

Art.  182. 

Page  210. 

3.  $78.61. 

4.  $25. 


Art.  186. 

Page  213. 

2.  $194.53. 

$2,470,228. 

$915.60. 

$128.16. 

$46.13. 

>^64.11. 

$554.40. 

$45.10. 

$432.64. 


11.  $210. 

12.  $158.60. 

14.  $4,416,494. 

15.  £179  15s.  M. 

nfar. 

17.  $517.50. 

18.  $4,795.56. 

19.  £3,166  136-. 

M. 

20.  $8,718.75. 

21.  $1,030.40. 

22.  $2,115.80. 

23.  $3,392.50. 

24.  $11,732.58. 

25.  $18,597.96. 

26.  $1,080. 

27.  $1,200. 

Art.  187. 

Pages  21Jr-217. 

$58,892. 

$28.35. 

$66  36. 

$1,387.80. 

$4,639.70. 

$2,955,767. 

$18.23. 

$2.65. 

$729,646. 

$44.50. 

$82.25. 


$393.42|. 


17.  $63.45. 

18.  $54,371. 

19.  $918.60. 

20.  $1,012.65. 

23.  $510.86. 

24.  $191,205. 

25.  $266.07. 

26.  $328.32. 

27.  $190,151. 

28.  $44,289. 

29.  $7.66-^. 
$55,417. 
$193.61. 
£52  10s. 
$79,875. 
$190,517. 
£12,170. 


38.  $372.60. 


39.  $4,606.87^. 

40.  $6,974.57*. 

41.  .If 843. 
$297.50. 
$3,875.37*. 
$964,219.'' 
$328. 
$74,025. 
$71,867. 
$532.50. 
$4,205,331. 
$5,385. 


Art.  188. 

Pages  217,218. 
2.  $0,526. 
3- 
4- 
5- 
6. 

7. 
10. 


$1,302. 


$1,449. 

$11,828. 
$7.92 ; 
.      $13.20. 

11.  $4.37;  $7.29. 

12.  $1,008; 
$2,176. 

13.  $1,408; 
$4,224. 

14.  $1,087; 
$1,521. 

Art.  189. 

Page  219. 

2.  $7.84. 

3.  $20.16. 

4.  $66.21. 

5.  $1,167. 

6.  $4,792. 

7.  $23,123. 


Art.  190. 

Page  219. 

I.  7%. 
4i%. 

7%. 
6%. 
7%. 
7%. 
5%. 


328 


ANSWEES. 


Art.  191. 


3  yrs. 
'3^  yrs. 
2  yrs. 
9  mos. 

1  yr.  2  mos. 

1  yr.  3  mos. 
171  yrs. 

2  yrs. 

4  yrs. 


Art.  192. 

Page  221. 

1.  $2,100. 

2.  $3,000. 
3-  '13,750. 

4.  $1,800. 

5.  $2,500. 

6.  $100. 

Miscellaneous. 

1.  $950. 

2.  7%. 

3.  5  yrs. 

4.  $225. 

5.  lyr-H.  A  mo. 

6.  $1,500. 

7.  2r'«- 

8.  14^.. 

9.  $646,333. 
10.  1  yrs.  11  mo. 

24:  da. 

Art.  19:i. 

Page  222, 

2.  $371.28. 

3.  $351,232. 

Art.  198. 

Pages  225, 220. 

2.  $252,074. 

3.  $200.55. 

5.  $128,152, 

6.  $5,081,628. 

7.  $474,887. 

8.  $700,452. 


10.  $405.44. 

Art.  200. 

Pages  229,231 

2.  $175,253. 

3.  $823,401. 

5.  $1,671.75. 

6.  $496.17. 


Art.  202. 

231, 23i 

$1,055.70. 

$10.62t. 

$30. 

$803.25. 
$494,491. 
$346. 87i. 
$4.95.    " 
$.59#. 


Art.  20s. 

Pages  233,234. 

3.  $1,454,546. 

$41.25. 

$66,364. 


$136. 

$1,090,909. 

$60,287. 

$680,581. 

$1,095,506. 

$9,830,425. 

$1,750. 

$4,000. 

$3,420,089. 

Art.  207- 

re  230. 


3.  $7,035. 

4.  $6,329. 

5.  $989.50. 

6.  $1,696,752. 

7.  $1,593. 

Art.  20s. 

Page  237. 
2.  $505,306. 


$765,853. 
$1,009,251. 


Art.  218. 

Pages  240, 241. 

6.  $2,090. 

7.  $105,322.50, 

8.  $168.75. 
9-  $112!%. 

10.  $10,125. 

11.  lllf%. 

12.  $8,000. 

15.  5fft^. 

16.  71%. 
18.  71|%. 

Art.  227. 

Page  24G. 

II.  $1,971.4U. 

Art.  229. 

Page  247. 

2.  $3,928.50. 

3.  $1,560.06. 

4.  $1,587.60. 

6.  £563  9.s'.8M 
8.  $6,057,143. 
9-  5,200 /r. 

Art.  23S. 

Page  250. 

5.  70  da. 

Art.  246. 

Pages  200, 261. 

2.  o. 

3.  5. 

4.  3. 
5-  \. 

6.  10. 

7.  30. 

9-  i. 

10.  I 

11.  28. 


12.  ,-v 

13.  2  lbs.  10  oz. 

14.  3  Ihs.  1  oz. 

15.  $3. 

16.  20  yds. 

Art.  250. 

Page  203. 

3.  20  yds. 

4.  27. 
5-  96. 

6.  9. 

7.  i 

8.  0.26. 

9.  $4.20. 
10.  £3i. 

II    $270. 83i. 

12.  $7.50. 

13.  £4  8."?.  6fZ. 

14.  3i  C. 

15.  $165.79. 

16.  49. 

17.  $5. 

18.  641. 

19.  $1.50. 

20.  $524. 

21.  $20,909. 

22.  6|  2/^s. 


Art. 

Pages 

3.  a'= 

4.  x= 

5.  a? 

6.  a?= 

7.  a'= 

8.  0?= 

9.  X— 

10.  a;= 

11.  a;= 

12.  X— 

13.  «-- 

14.  x= 

15.  0!= 

16.  «!= 

17.  X— 

18.  a;= 

19.  X— 

20.  0?=: 


205-208. 

643  J  W2I. 
58.75. 
$125,797. 
$190.85. 
28. 

$3.29. 
11. 
300. 

86/0%  da. 
l^  yds. 
48  men. 
11  yds. 

$3. 
20  hi(. 
$8.22.i. 
1(\%  da. 
$9:95. 


ANSWERS. 


329 


21.  a;=%ld2. 

22.  a'=112|  hii. 

23.  x=3d.l5fr. 

24.  ir=35  7«. 

25.  ir=$820. 

26.  X =162  da. 

27.  .T=$3,597.2'2|. 

28.  »•= $9,972.9:^%, 

29.  a;=S7,200. 
32.  a;=10^ws€s, 

^»f.  253. 

Page  269. 

5.  A.,  $180; 
B.,  $85 ; 
C,  $240. 

6.  A.,  $37-1  ; 
B.,  $621. 

7.  A.,  $130; 

B,  $195 ; 

C,  $325. 

8.  A.,  $150; 
B.,  $225 ; 
C,  $270  ; 
D.,  $375. 

Art,  258. 

Page  276, 

1.  64. 

2.  625. 

3.  2,744. 

4.  625. 

5.  9,604. 

6.  .0081. 

7.  .0225. 

8.  15,625. 

9.  .0359. 
10.  11.56. 


II. 


(J4- 


12.  tf- 

13.  15|. 

14.  34fi. 

15.  410,V 

Art,  261, 

Page  ooo- 
2.  98. 


3.  llo. 

4.  585. 

5.  972. 

6.  131. 

7.  194. 

8.  204. 

9.  229. 

10.  M. 

11.  ^^.. 


Miscellaneous 
Examples. 

1.  369. 

2.  15. 

3.  2.  2.  2.  5.  5 

7.7. 

4.  2.  3.  5.  7.  19, 

5.  458. 

6.  35. 

7.  1,800. 

8.  3,556. 

9-  '^m- 
10.  11. 

TT        11 

12.  l-V- 

13.  34.639. 

14.  1.2002. 

15.  $240. 

16.  480. 

17.  A.,  $51 ; 
B.,  $127-J-  ; 
C,  $1781 

18.  lUda. 

19.  ^Wihu.^pks. 

4:qts. 

20.  12  bu. 

21.  IShfj  Ct8. 

22.  $250 ;  $350 ; 
$400. 

23.  $273. 

24.  10  mi. 

25.  47,250  lbs. 

26.  60  da. 

27.  $4,070. 

28.  $160. 

29.  $4,350. 

30.  $898.80. 


18%. 
25%. 
40%. 
53  mi. 
S15. 
120  rds. 
195  bu. 
30  men. 
$830. 
214  sheep. 
25,920  men. 
250  melons. 
216  ft. 
S.,  $660 ; 
J.,  $577.50  ; 
B.,  $412.50. 
$2,604i. 
$4,102fi 
18  ba^es. 
2|  mo. 
$36.75. 
$131.35. 
1st,  $15  ; 
2d,  $22.50  ; 
3d,  $7.50. 
n,66^id.ft. 
21  a  l3i  cu. 

ft- 

A.,  $128; 
B.,  $168; 
C,  $104. 
$1  per  yd. 
50  ft. 
$32,000. 
25%. 
2,400 
9  iceeks. 
3,468  «g./^. 
Son,  $8,000;! 
eldest    dmi., 

$6,000; 
younger  dav. 

$4,000. 
286,688. 
120  men. 
14. 
5  hrs.    27-i\ 

min. 
750  men. 


68. 
69. 
70. 

71. 

72. 

73. 
74- 
75- 
76. 

77. 

78. 

79. 
80. 


66  yrs. 
28f  da. 
12  hrs.  10  m. 

P.M. 

125  reams. 
12  da. 
75  cu.  yds. 
11  da. 
23^. 


82. 


83. 


85. 

86. 

87. 


94. 

95. 
96. 


97. 
98. 
99. 
100. 


15.76  m. 
167.4. 

39.06  Mlog. 
Man  6. 15  fr.; 
worn.  2.05  fr. 
A.,  $380  ; 
B.,  $430. 
W.,  $4,420. 
B.,  $3,080. 
A.,  $4,080  ; 
B.,  $3,100  ; 
C,  $1,930. 
A.,  $7^19.84; 
B.,  $5,086. 70-f\; 
C,  |5,(43,05-,\. 
5  better, 

4  poorer, 
16i  da. 
Elder,$5.A00; 

younger, 
$4,600. 
$1 ,500. 
$4,800. 
16  men. 
72  in. 
B.,  $3,750 ; 
C,  $3,125. 
Capt.,  $243; 
man,  $162;   , 
boy,  $54. 
$l,666f. 
190. 

A.,  $445 ; 
B.,$230; 
C,  $335. 

lO^Wiy. 

5  hrs.  20  m. 
45  yrs. 

65. 


YB  35831 


i  iO. 


54)475 


UNIVERSITY  OF  CAUFORNIA  LIBRARY 


